Two questions asked

1. Does God pray?
Jewish writings says: “pray as if everything depended upon G-d, but act as if everything depended upon ourselves.”
Isaiah (56:7). The prophet offers a universal vision of foreigners, not just Jews, coming to worship at G-d’s holy mountain. G-d says: “I will let them rejoice in My house of prayer.” In Hebrew, though, that last phrase, “My house of prayer,” literally reads, “the house of My prayer.” In other words, the Temple may be viewed, not only as the place where G-d is worshiped, but also as G-d’s very own synagogue, where G-d goes to pray.
G-d prays: ‘May it be My will that My mercy may suppress My anger, . . . so that I may deal with My children in the attribute of mercy and, on their behalf, stop short of the limit of strict justice.’ In the Talmud story of having entered the Holy of Holies, the most sacred enclosure of the ancient Temple, on Yom Kippur, the holiest day of the year. There, the Rabbi sees G-d, enthroned on High, crowned in glory. G-d appeals to the High Priest: “‘Bless Me, My son.’ Rabbi then blesses the L-rd with the same words of G-d’s own prayer: ‘May it be Thy will that Thy mercy may suppress Thy anger, . . . so that Thou may deal with Thy children according to the attribute of mercy and may, on their behalf, stop short of the limit of strict justice!’ Rabbi further asserts that, after the blessing, G-d nodded to him, approving of the prayer!
This teaching is profound, for it suggests that the prayerful words of a human being may have a profound effect on
G-d. Our prayers do matter to G-d. Our petitions may move the L-rd to action. Indeed, our supplications may inspire G-d to pray, just as we pray. Perhaps G-d can bring water to parched fields, can find a way to mend a broken bridge, can work to rebuild a ruined city. At the very least, G-d may add Divine prayers to ours, with faith that our thirst will be slaked, our spirits will be healed, our nations may find peace, in our time of distress, G-d lovingly picks us up, and kisses us.
May we ever pray with faith. May we live with trust in the L-rd, Who loves us, cares for us, and only wants goodness for us. Let us not see our misfortunes as evidence that G-d does not exist, or is callous to our needs. Instead, through each of our trials and tribulations, may we hear G-d’s prayer for us. And may G-d hear our prayers for G-d.
2. Are we to pray 'in the name of' the God of Abraham, Isaac and Jacob?
Now that is a trick question - if you are a 'Jesus only' you would say 'that is the name' but Scripture says otherwise: "From the rising of the sun to its setting," says the prophet Malachi (1:11) in an oracle of the L-rd, "My name is great among the nations, and in every place incense is offered to My name, and a pure offering; for My name is great among the nations, says the L-RD of hosts." For almost 2,000 years, at least some Christian thinkers have taken Malachi’s oracle to mean that wherever genuine sacrifice is offered, it is made to YHWH, the one true
G-d.
In everything Y'Shua did He pointed back to the Father. All songs were sang to the Father and prayers directed to Him for there is only one G-d (Deut. 6:4-9). Jn. 10:25 ...the works that I do in My Father's name, they bear witness of Me.
But it also says to ask in Y'Shua's name, since He is the only son of G-d and our advocator: Jn. 15:16 ...that whatsoever ye shall ask of the Father in my name, He may give it you.
Bottom line: I personally pray to the Father and end it "in Y'Shua's name."