In whose name should we ask of God the graces we stand in need of?
We should ask of God the graces we stand in need of in the Name of Jesus Christ, as He Himself has taught us and as is done by the Church, which always ends her prayers with these words: Through our Lord Jesus Christ.
Why should we beg graces of God in the Name of Jesus Christ?
We should beg graces of God in the Name of Jesus Christ because He is our Mediator, and it is through Him alone that we can approach the throne of God. (Catechism of Saint Pius X, Prayer in General, no. 8-9)

Prayer directed to God should be united with Christ, the Lord of all men, the one mediator, through whom alone we have access to God. Christ so unites the whole human family to himself that there is an intimate and necessary relationship between the prayer of Christ and the prayer of the whole human race. For in Christ alone human religion achieves its redemptive value and purpose. A close and special bond exists between Christ and those whom, through the sacrament of regeneration, he makes members of his body, the Church. All the riches belonging to the Son flow from him as from the head into the whole body: the pouring out of the Spirit, truth, life and a share in his divine sonship, which he revealed to us in all his prayer on earth. The whole body of the Church shares in the priesthood of Christ. The baptized, by regeneration and the anointing of the Holy Spirit, are consecrated into a spiritual house and a holy priesthood. They become capable of taking part in the worship of the New Testament, not thanks to themselves, but to the gift and merits of Christ. […] Christian prayer draws its dignity from its sharing in the filial relationship of the Only-begotten Son to the Father. The prayer he expressed in his earthly life with his own words in the name of and for the salvation of the entire human race, he continues to address to his Father in the whole Church and in all her members. (Congregation for the Divine Worship and the Discipline of the Sacraments. General Introduction on the Liturgy of the Hours, The Importance of the Liturgy of the Hours or the Divine Office in the Life of the Church, ch. 1, no. 6-7, February 2, 1971, English translation 1974)

Prayer is the living relationship of the children of God with their Father who is good beyond measure, with his Son Jesus Christ and with the Holy Spirit. […] Thus, the life of prayer is the habit of being in the presence of the thrice-holy God and in communion with him. This communion of life is always possible because, through Baptism, we have already been united with Christ (Rom 6:5). Prayer is Christian insofar as it is communion with Christ and extends throughout the Church, which is his Body. Its dimensions are those of Christ’s love (Eph 3:18-21). (Catechism of the Catholic Church, no. 2565)

This is why when a Christian prays, even if he is alone, his prayer is in fact always within the framework of the ‘Communion of Saints’ in which and with which he prays, whether in a public and liturgical way or in a private manner. Consequently, it must always be offered within the authentic spirit of the Church at prayer, and therefore under its guidance, which can sometimes take a concrete form in terms of a proven spiritual direction. The Christian, even when he is alone and prays in secret, is conscious that he always prays for the good of the Church in union with Christ, in the Holy Spirit and together with all the Saints. (Congregation for the Doctrine of the Faith. Letter to the Bishops on some aspects of Christian meditation, no. 7, October 15, 1989)

For the Communion of Saints, as everyone knows, is nothing but the mutual communication of help, expiation, prayers, blessings, among all the faithful, who, whether they have already attained to the heavenly country, or are detained in the purgatorial fire, or are yet exiles here on earth, all enjoy the common franchise of that city whereof Christ is the head, and the constitution is charity. (Leo XIII. Encyclical Mirae Caritatis, May 28, 1902)

But He holds out a ratification not only of sentences of excommunication, but of every petition which is offered by men holding together in the unity of the Church; for He adds, ‘Again I say unto you, that if two of you shall agree upon earth,’ whether in admitting a penitent, or casting out a forward person, ‘touching anything which they shall ask,’ anything, that is, that is not against the unity of the Church, ‘it shall be done for them by my Father which is in heaven.’ By saying, ‘which is in heaven,’ He points Him out as above all, and therefore able to fulfill all that shall be asked of Him. Or, He is in the heavens, that is, with saints, proof enough that whatever worthy thing they shall ask shall be done unto them, because they have with them Him of whom they ask. For this cause is the sentence of those that agree together ratified, because God dwells in them, ‘For where two or more are gathered together in my name, there am I in the midst of them.’ (Saint Thomas Aquinas. Catena Aurea 4818, Matthew 18:18-20)

The unity of the praying Church is brought about by the Holy Spirit, the same Spirit who is in Christ, in the whole Church, and in each baptized person. ‘This Spirit comes to help us in our weakness’ and ‘expresses our plea in a way that could never be put into words’ (Rom 8:26). As the Spirit of the Son, he breathes into us ‘the spirit of adopted sons, and makes us cry out, ‘Abba, Father!’’ (Rom 8:15; cf Gal 4:6; 1Cor 12:3; Eph 5:18; Jude 20). There can be no Christian prayer without the action of the Holy Spirit. He unites the whole Church and leads us through the Son to the Father. (Congregation for the Divine Worship and the Discipline of the Sacraments. General Introduction on the Liturgy of the Hours, The Importance of the Liturgy of the Hours or the Divine Office in the Life of the Church, Ch.1, no. 6-7, February 2, 1971, English translation 1974)

The Holy Ghost is the author of our prayers; and under His guiding influence, we cannot fail to be heard. We have received the spirit of adoption of sons, whereby we cry, ‘Abba, (Father)’. This spirit succours our infirmity and enlightens our ignorance in the discharge of the duty of prayer; nay, even, as the Apostle says, He asketh for us with unspeakable groanings. (Catechism of Trent. 4000, IV, Motives of Confidence in Prayer, pg. 296)

The manner of praying is also a matter of the highest moment. Though prayer in itself is good and salutary, yet if not performed in a proper manner it is unavailing. Often we do not obtain what we ask, because, in the words of Saint James, we ask amiss. […] We must, then pray in spirit and in truth; for the heavenly Father seeks those who adore Him in spirit and in truth. He prays in this manner whose prayer proceeds from an interior and intense ardour of soul. This spiritual manner of praying does not exclude the use of vocal prayer. Nevertheless, that prayer which is the vehement outpouring of the soul, deservedly holds the first place; and although not uttered with the lips, it is heard by God to whom the secrets of hearts are open. […] This practice of praying in spirit is peculiar to Christians, and is not at all used by infidels. Of these Christ the Lord has said: When you pray, speak not much, as the heathens; for they think that in their much speaking they may be heard. Be not ye, therefore, like to them, for your Father knoweth what is needful for you before you ask him. […] Neither do the prayers of the hypocrite proceed from the heart; and against the imitation of their example, Christ the Lord warns us in these words: When ye pray, ye shall not be as the hypocrites that love to stand and pray in the synagogues, and corners of the streets, that they may be seen by men. Amen I say, to you they have received their reward. But thou, when thou shalt pray, enter into thy chamber, and having shut the door, pray to thy Father in secret; and thy Father who seeth in secret will repay thee. Here the word chamber may be understood to mean the human heart, which we should not only enter, but should also close against every distraction from without that could deprive our prayer of its perfection. For then will our heavenly Father, who sees perfectly our hearts and secret thoughts, grant our petitions. (Catechism of Trent. 4000, IV, The Lord’s Prayer, pg. 298)

Do all the children of the Church share in this communion of goods?
All Christians who are in the grace of God share in the communion of internal goods, while those who are in mortal sin do not participate in these goods.
Why do not those who are in mortal sin participate in these goods?
Because that which unites the faithful with God, and with Jesus Christ as His living members, rendering them capable of performing meritorious works for life eternal, is the grace of God which is the supernatural life of the soul; and hence as those who are in mortal sin are without the grace of God, they are excluded from perfect communion in spiritual goods, nor can they accomplish works meritorious towards life eternal. […] Who are they who do not belong to the Communion of Saints?
Those who are damned do not belong to the Communion of Saints in the other life; and in this life those who belong neither to the body nor to the soul of the Church, that is, those who are in mortal sin, and who are outside the true Church.
Who are they who are outside the true Church?
A. Outside the true Church are: Infidels, Jews, heretics, apostates, schismatics, and the excommunicated. (Catechism of Saint Pius X. no. 4-5.10-11)

The advantages of so many and such exalted blessings bestowed by Almighty God are enjoyed by those who lead a Christian life in charity, and are just and beloved of God. As to the dead members; that is, those who are bound in the thraldom of sin and estranged from the grace of God, they are not so deprived of these advantages as to cease to be members of this body; but since they are dead members, they do not share in the spiritual fruit which is communicated to the just and pious. (Catechism of Trent, 1090, IX Part 2, The Communion of Saints, pg. 84)

[Errors of Martin Luther] Excommunications are only external penalties and they do not deprive man of the common spiritual prayers of the Church. […] [Censure of the Supreme Pontiff] we condemn, reprobate, and reject completely each of these theses or errors as either heretical, scandalous, false, offensive to pious ears or seductive of simple minds, and against Catholic truth. (Leo X. Bull Exurge Domine, June 15, 1520)

What is the first and best disposition to render our prayers efficacious?
The first and best disposition to render our prayers efficacious is to be in the state of grace; or if we are not in that state, to desire to put ourselves in it. (Catechism of Saint Pius X, Prayer, no. 14)

Neither prayer nor any other virtuous act is meritorious without sanctifying grace. And yet even that prayer which impetrates sanctifying grace proceeds from some grace, as from a gratuitous gift, since the very act of praying is ‘a gift of God,’ as Augustine states (De Persever. 23). (Saint Thomas Aquinas. Summa Theologica, II-II, q. 83, a. 15)

Where does prayer come from? Whether prayer is expressed in words or gestures, it is the whole man who prays. But in naming the source of prayer, Scripture speaks sometimes of the soul or the spirit, but most often of the heart (more than a thousand times). According to Scripture, it is the heart that prays. If our heart is far from God, the words of prayer are in vain. […] Christian prayer is a covenant relationship between God and man in Christ. It is the action of God and of man, springing forth from both the Holy Spirit and ourselves, wholly directed to the Father, in union with the human will of the Son of God made man. (Catechism of the Catholic Church, nos. 2562.2564)

For when the Lord urged unanimity and peace upon His disciples, He said: ‘I say to you that if two of you agree upon earth concerning anything whatsoever that you shall ask, it shall be granted you by my Father who is in heaven. For wherever two or three have gathered together in my name, I am with them, ‘showing that the most is granted not to the multitude but to the unanimity of those that pray. […] but rebuking the faithless for their discord and commanding peace to the faithful by His word, He shows that He is with two or three who pray with one mind rather than with a great many who are in disagreement, and that more can be obtained by the harmonious prayer of a few than by the discordant supplication of many. […] The Lord, moreover, is speaking of His Church, and He is speaking to those who are in the Church, that if they are in agreement […] Where there are two or three, I am with them,’ He who established and made the Church did not separate men from the Church, but rebuking the faithless for their discord and commanding peace to the faithful by His word, He shows that He is with two or three who pray with one mind rather than with a great many who are in disagreement, and that more can be obtained by the harmonious prayer of a few than by the discordant supplication of many. (Saint Cyprian of Carthage. The Unity of the Catholic Church, XII, PL 4, 508-509)

So the twelfth degree [of pride] may be called the habit of sinning, because in it the fear of God is lost, and its place is taken by scorn. To what extent may prayer be offered for the incorrigible, and spiritually dead? For such an one, says John the Apostle, I do not say that any one shall pray. But sayest thou, O Apostle, that no one may Hope? Surely he who loves that man may groan. He ventures not to pray, he need not for-bear to weep. What is this that I say that perchance there remains the resource of hope, where prayer has no place? Take an instance of one who believes and hopes, yet does not pray. Lord, she says, if thou hadst been here, my brother had not died. […] Wilt thou show wonders to the dead ? or shall physicians raise to life and give praise to thee? And concerning him who has been dead. Four days. Shall anyone in the sepulcher declare thy mercy; and thy truth in destruction? Meanwhile it is possible that the Saviour may be pleased to meet us unforeseen and unexpectedly, and moved by the tears, not by tried prayers of the bearers, to restore the dead man to those who live, or actually to recall from among the dead one who is already buried. But I should describe as dead the man who by excusing his sins, has already come down to the eighth degree. For praise perisheth from the dead as from one who does not exist. But after the tenth degree, which is third from the eighth, he is already being carried out into liberty to sin, when he is expelled from the monastic community. But after he has passed the fourth degree he is rightly said to be ‘four days dead’, and when he falls into the fifth degree of habitual sin he is already buried. They should nevertheless realize the great danger which those incur whom the Church […] dares not to mention in her worship. For when on Good Friday prayer is expressly offered for certain wicked persons, no mention is made of those who are excommunicated. (Saint Bernard of Clairvaux. The Twelve degrees of humility and pride, Ch. 21, no. 51-52.55-56)

The last degree [of prayer] is that of those who not only do not repent of their sins and enormities, but, adding crime to crime, dare frequently to ask pardon of God for those sins, in which they are resolved to continue. With such dispositions they would not presume to ask pardon from their fellow­man. The prayer of such sinners is not heard by God. It is recorded of Antiochus: Then this wicked man prayed to the Lord, of whom he was not to obtain mercy. (Catechism of Trent. 4000, Part IV, The Prayer of Unbelievers)

Miserable the sinner that hardens his heart and resists the divine calls: His heart shall be as hard as a stone and as firm as a smith’s anvil (Job 41:15). Instead of yielding to the graces and inspirations of God, and being softened by them, the unhappy man becomes more obdurate, as the anvil is hardened by repeated strokes of the hammer. In punishment of his resistance to the divine calls, he will find his heart in the same miserable state at the very hour of death, at the moment of passing into eternity. A hard heart shall fare evil at the last. Sinners, says the Lord, you have, for the love of creatures, turned your back upon me. They have turned their back upon me, and not their face; and in the time of their affliction they will say: Arise, and deliver us. Where are the gods thou hast made thee? Let them arise and deliver thee (Jer 2: 27). They will have recourse to God at death; but he will say to them: Why do you invoke me now? Call on creatures to assist you; for they have been your gods. The Lord will address them in this manner, because, in seeking him, they do not sincerely wish to be converted. Saint Jerome says that he holds, and that he has learned from experience, that they who have to the end led a bad life, will not die a good death […] It is a marvellous thing that God unceasingly threatens sinners with an unhappy death. Then they shall call upon me, and I will not hear (Prov. 1:28).Will God hear his cry when distress shall come upon him? (Job 27: 9). I also will laugh in your destruction, and will mock (Prov 1:26). According to Saint Gregory, God laughs when he is unwilling to show mercy (Mor 1, 9, c. 20). Revenge is mine, and I will repay them in due time: (Deut 22: 35). The Lord pronounces the same threats in so many other places: and sinners live in peace as securely as if God had certainly promised to give them, at death, pardon and paradise. It is true that at whatsoever hour the sinner is converted God promises to pardon him. But he has not promised that sinners will be converted at death: on the contrary, he has often protested that they who live in sin shall die in sin. You shall die in your sins (Jn 8:21-24). He has declared that they who seek him at death shall not find him. You shall seek me, and shall not find me (Jn 7:34). (Saint Alphonsus Liguori. Preparation for Death, VI, 2-3)