Did Peter ever ask Jesus for forgiveness?

Mankind cannot avoid being forgiven by their own action or inaction

You ask

Why would Jesus ever give his role as shepherd to someone like this?

Because Jesus is more forgiving than you are, and doubtless more forgiving than I am. Your question includes a wide variety of signs and behaviors that, in your opinion, are preconditions for forgiveness. Forgiveness doesn't require any precondition.

God's love is unconditional, as is the love Jesus has for us. We can be forgiven without being or acting repentant.

• That said, in Catholic tradition, for example, remission of sins and absolution is considered to be incomplete without sincere repentance on the part of the sinner when confession mortal sins and seeking to turn back toward God, toward Christ.

  But none of that restricts God', nor Christ's, ability to forgive. Put another way, no action of a sinner can prevent divine forgiveness. We don't have the power.

  Example: I can forgive my wife, or my enemy, or as Pope John Paul II did I can forgive someone who tried to shoot and kill me. Forgiveness is a gift given that is In No Way contingent upon the actions of the recipient.

The problem to solve in your question isn't Peter being forgiven; it is in your misunderstanding of how forgiveness works. It's a common misunderstanding of more than forgiveness.

Some things can only be given

Thanks can only be given. I can't take your thanks.
An apology can only be given. You can't take an apology from me.
Love can only be given. You can't take love from someone.
Likewise, forgiveness. It can only be given. You can't stop me from forgiving you, but you could, for example, not accept an apology. (but I'd still have offered or given it)

About Jesus' forgiveness, and his unconditional giving

With Jesus dying on the Cross, he gave all of his humanity for the forgiveness of all of mankind (not just me, not just you, and not just Peter). Jesus' forgiveness is of a divine quality. It didn't matter what Peter did, or did not, do. Jesus' forgave, and forgives, him. And you. And Sola Gratia. And me.

Luke 23:34

Father, forgive them, for they know not what they do

This answer was a challenge to the frame of the question.

John 21:15-19 is where Jesus asks Peter three times, "Lovest thou me?" To understand what is going on in this exchange, you need to know the context.

In Matthew 26:33, before Peter denied Jesus, he basically said that he loved Jesus more than anyone else. "Though all men shall be offended because of thee, yet will I never be offended. I would die before I would deny you."

So, after Peter's thrice denial, meeting Jesus again, he is waiting for the "I told you so." Jesus asks first, "Do you still think you love (agape) me more than anyone else?" Agape is the love where you lay down your life for another. Peter's sheepish reply in effect says, "I would characterize my love more like brotherly love." I've been busted. I'm embarrassed. Don't rub it in. Can we move on?

So Jesus changes the question a little: "Do you agape me? Maybe not more than everyone else, but still, do you agape me?" Peter is not going to be boastful again, so he sticks with the brotherly love, having already proved that his love was not the agape variety.

Then Jesus changes the question again: "So you would characterize your love as brotherly love?" Peter, wishing Jesus would move on and not press the point, agrees: "Yes, my love for you is brotherly love."

At that point Jesus says, Peter, you are once again mistaken. Your love is agape love. Jesus then proceeded to tell Peter how he would lay down his life to the glory of God.

Thank God He does not write us off after we make a mistake(s).

Yes, Peter did repent, which Jesus calls "turning back" (from his betrayal):

Luke 22:31-32 (DRB) And the Lord said: Simon, Simon, behold Satan hath desired to have you [plural], that he may sift you as wheat: But I have prayed for thee, that thy faith fail not: and thou, being once converted, confirm thy brethren.

The Greek ποτε επιστρεψας (here translated "being once converted") means "when thou shall have turned back." Strongs's Definition of ἐπιστρέφω: "to revert (literally, figuratively or morally)."

There is no physical turning in context to be had. This must refer to his betrayal and subsequent repentance.