According to Trinitarians, who is the begotten "son" in Psalm 2 and what happened on this "day"?
As a trinitarian myself I believe, along with others, the "day" ("This day have I begotten you") is the Day of Christ's Resurrection from the dead when his dead human body was raised to life and given an immortal, resurrection body to reign forever as the God-man, in accordance with Acts 13:33.
"This day have I begotten thee" does not refer to a day at the beginning of eternity nor to his being "eternally begotten".
There are (at least) three quotes of Psalm 2:7 in the NT:-
In Acts 13:33, the emphasis is on the words "this day". Acts 13:28-34 reads:
28 And though they found in him no guilt worthy of death, they asked Pilate to have him executed. 29 And when they had carried out all that was written of him, they took him down from the tree and laid him in a tomb. 30 But God raised him from the dead, 31 and for many days he appeared to those who had come up with him from Galilee to Jerusalem, who are now his witnesses to the people. 32 And we bring you the good news that what God promised to the fathers, 33 this he has fulfilled to us their children by raising Jesus, as also it is written in the second Psalm, “‘You are my Son, today I have begotten you.’ 34 And as for the fact that he raised him from the dead, no more to return to corruption, he has spoken in this way, “‘I will give you the holy and sure blessings of David.’ (ESV)
Now we have a clearer idea from Acts 13:28-34 of what is being referred to by the word "begotten" in Psalm 2:7, we can get yet more detail from the phrase in Revelation 1:5,
"the firstborn of the dead".
This shows that "begotten" in Psalm 2:7 is referring to Christ's being raised to life from the dead. Rev 1:5 confirms the resurrection of our Lord is being referred to. He is the first to be raised from the dead with the immortal, resurrection body.
(By the way, notice the way the NIV uses dynamic equivalence and I think strays too far from the Greek in the following verses Acts 13:33, Hebrews 1:5, Hebrews 5:5, giving the wrong sense. "Today I have become your Father" leads away from the idea of "Today I have begotten you", i.e. risen you to new life (from the dead).)
In Hebrews 1:5 the emphasis is on the word "son". And who is this "son"? Hebrews 1:5-8 says:
5 For to which of the angels did God ever say, “You are my Son, today I have begotten you”? Or again, “I will be to him a father, and he shall be to me a son”? 6 And again, when he brings the firstborn into the world, he says, “Let all God's angels worship him.” 7 Of the angels he says, “He makes his angels winds, and his ministers a flame of fire.” 8 But of the Son he says, “Your throne, O God, is forever and ever, the scepter of uprightness is the scepter of your kingdom. (ESV)
And in Hebrews 5:5, the emphasis is that the man Jesus Christ did not choose the honour of being the Son of God, but God chose him. This choosing is in the context of our Lord Jesus becoming the High Priest: the man Jesus Christ did not choose to be the eternal High Priest, but it was God's choice:
4 And no one takes this honour for himself, but only when called by God, just as Aaron was. 5 So also Christ did not exalt himself to be made a high priest, but was appointed by him who said to him, “You are my Son, today I have begotten you” (Heb 5:4-5, ESV).
So the word "son" refers to our Lord Jesus Christ who is so far superior to the angels that they must worship him and, according to Hebrews 1:8, "God".
To be sure to answer all your other questions directly:
What day was this "son" begotten?
He was begotten on the day of his resurrection.
Can we assume this "son" is Jesus?
No assumptions are needed: it is clear from Hebrews ch1, ch5 and Acts 13-28-34 that Psalm 2 refers to our Lord Jesus Christ.
Was this the "day" YHWH also enthroned him in v16?
Uuh? Verse 16 of what passage?
Can YHWH beget Himself? Do fathers ever beget themselves? (If this son is Jesus and Jesus is YHWH)
YHWH is a term for the triune God. Sometimes in scripture the term refers to the Father, sometimes to the Holy Spirit, sometimes to Jesus Christ the Son, and sometimes to all the persons in the Godhead.
I believe the idea of begetting is not very useful when referring to the divine nature of Christ because it suggests there was a time when Christ did not exist, and saying he is "eternally begotten" is a contradiction in terms. Quoting Dr Robert Reymond in "A New Systematic Theology of the Christian Faith", page 329,
He [John Calvin] concludes his treatment of the doctrine of the Trinity by declaring the ancient doctrine of "eternal generation" to be of "little profit", unnecessarily "burdensome", "useless trouble", and "foolish":
'While I [says Calvin] am zealous for the edification of the church, I felt that I would be better advised not to touch upon many things that would profit but little, and would burden my readers with useless trouble. For what is the point in disputing whether the Father always begets? Indeed, it is foolish to imagine a continuous act of begetting, since it is clear that three persons have subsisted in God from eternity.' (Calvin's Institutes book I, chapter xiii, 29).
The New International Version chooses "the one and only" in places where the KJV uses "only begotten". The idea in John 3:16 and other places in the NT is that of uniqueness. Jesus is "the one and only Son of God", in a way that no other is or ever can be. The Scriptures are clear, "God was manifest in the flesh", 1 Tim 3:16. Those who use the term "begotten" in John 3:16 and the like passages want to emphasise that Jesus is of the same nature as the Father, i.e. that Jesus is God just as much as the Father is God. But this idea of equality is clear from Jesus calling God his Father (see John 5:18).. every time he called God his Father he was claiming to have the same nature as God the Father. Introducing the idea of the Son of God being "eternally begotten" to prove his deity is foreign to the NT.
In Psalm 2 the emphasis seems to be on the promotion of Christ's human nature to an immortal body in order to reign forever in glory, i.e. the receipt of a resurrection body. Now a human is reigning in glory, as the God-man.
Did Jesus beget a son? (If this son is NOT Jesus and Jesus is YHWH).
No. Isaiah 53:8(?).