My own translation of 1 John 2:12-14:
12 I am writing to you little children, because through his name your sins have been forgiven. 13 I am writing to you fathers because you have known Him(a) from the beginning. I am writing to you young men because you have subdued the evil one.
I have written to you little children because you have known the Father.
14I have written to you fathers because you have known Him(a) from the beginning. I have written to you young men because you are strong men and the word of God abides in you, for you have subdued the evil one.
(a) properly, "the One"
Anything that is suggested about why this change of tense occurs, can only be conjecture.
My guess is, the letter was sent in two parts. The "I am writing..." sentences were the the last things written in the first part, and the "I have written..." sentences are the first things written in the second part — the writer having taken up from where he left off. Of course, he wouldn't have had a copy of the first, but he would remember the essence of how he finished it.
As possible evidence of this, τεκνία is the word for "little children" in the first part, but παιδία is used in the second. I think it unlikely that the writer would use different words in such close proximity if the epistle were continuous.
With regard to the "new commandment", I think a hint of it is given in verse 12, "because through his name your sins have been forgiven", but the full statement of it is not given until 1 John 3:23, which I would give like this:
The "new commandment" is not really new, but the writer of the epistle is emphasizing the importance of the NAME of Jesus Christ. For example, if the brothers cared about the name of their Saviour, then they wouldn't bring shame upon it by boasting of being in the light while at the same time expressing hatred towards one another (1 John 2:9-11). The world is looking for reasons not to put faith in the name of Jesus, and those who behave in such a way give them greater cause not to.
"Loving one another" is unarguably the principle theme of this epistle. Chapter 4 in particular, where more than half of the verses contain the word "love". By way of example here is my translation of 1 John 4:8-11:
7 Dear ones! We should love one another, because love is from God. And all who love have been begotten from God, and know God. 8 He who does not love has not known God, because God is love.
9 The love of God was manifest in us in this way: God sent his son ‒ the only one ‒ into the world so that we might live through him.
10 Herein is love: not that we loved God, but that he, himself, loved us and sent forth his son ‒ an atonement for our sins.
My translation of 1 John 2:7-8:
7 Dear ones! I am not writing a new commandment to you, but an old commandment that you have had from the beginning. The commandment, the old one, is the word that you heard. 8 Then again (πάλιν), I am writing a new commandment to you, one that is true in him and in you, because the darkness has passed, and the light, the true light, is shining even now.
The word πάλιν (Strong's G3825) at the beginning of verse 8, is not being used by the author to indicate that what follows is a "reiteration" of something he has said before, but to indicate that he has had a "second thought" about what he has just written.
I can't help being moved to see a connection between the author's emphasis on the behaviour of those who make claim to the "name of Jesus" and the injunction of the third commandment:
Exodus 20:7 (KJV)
7 Thou shalt not take the name of the Lord thy God in vain; for the Lord will not hold him guiltless that taketh his name in vain.
For the "name of Jesus" to stand out above all the rest, the behaviour of the brothers must be in keeping with the message of love in the Gospel, otherwise Jesus' name will mean "nothing", just another vanity -- an empty/worthless thing.
It is duplicitous for one's words to preach enlightenment and one's actions to preach something different. One who behaves in such a way WILL be seen for what they are, but the real damage is not done to them, but to "the name of Jesus". Jesus' name ends up becoming just another means of fooling the people for the purpose of exploiting them.
The bulk of what the author has to say is gentle and loving and uplifting as he, himself, cannot help but model the message of the words, but interspersed in the text are clear statements of what duplicitous behaviour looks like. For example my translations of 1 John 1:6 and 10:
6 If we were to say we have communion with him, but were to walk in darkness, then we have lied, for we do not do what is true...If we were to say we have not sinned, then we make him a liar. So, his word is not in us.
And 1 John 2:4,22:
He who says, "I have known him!" even while not keeping his commandments, is a liar, and the truth is not in him.... Who is the liar, if not he who has denied, as though Jesus is not the Christ. This is he who is the antichrist ‒ the one who has denied the Father and the Son.
The author initially said he wasn't writing a new commandment, but when he gave it a second thought it stuck him that he was, indeed, giving his readers something new to think about in regard to the old commandment (John 13:34, John 15:12,17).