The answer to both of your questions is that it's up to how you interpret. Hero doesn't give us a clear reason for why she forgives Claudio: in fact, the whole forgiveness thing is slightly vague and interpretable at the very end of the play:
Give me your hand before this holy friar. I am your
husband if you like of me.
And when I lived I was your other wife; [Unmasks.]
And when you loved you were my other husband.
One Hero died defiled; but I do live,
And surely as I live, I am a maid.
Hero never addresses Claudio beyond that. And she doesn't say "I forgive you". She reiterates the fact that she is a virgin ("maid"), and hasn't had sex as Claudio had suggested. So is this a forgiveness?
You can play it both ways, actually - as if, because she's gone along with the plan, she must want to forgive Claudio and become remarried. You can also suggest that she's wounded, hurt, and slightly disturbed by everything that has happened... and that she doesn't really know what to say.
Who knows? As for how their marriage will turn out, well, I think Shakespeare deliberately leaves that one open. You have a couple who argue with each other, tease each other, and are prepared to have a good fight (Beatrice and Benedick). And you have a supposedly perfect couple, who seem to communicate without words but with bashful looks, who have now come out of the other side of tragedy. Who'll be happy? Who'll last longer?
My money would be on Beatrice and Benedick. But Shakespeare lives it wide open.
What I have argued before in the past, Claudio's and Hero's love isn't "Real love" they fell in love by other's looks. While Beatrice and Benedick have fallen for each other's wit and humor.
So, Hero, being silly old Hero, didn't forgive him, but, didn't punish him. Shakespear did this to confuse us, and I love it much enough. :)
With Love and Care,