Mother's day is an easy one, will keep it all about the kids. Her birthday though will be a bot trickier. I guess some kind of simple gift, not too personal but something that lets her know that i care about her. Something thoughtful? Or should it just be a card?
I would skip the gifts. It is pursuing and will drive her further away.
Yes the kids can give gifts and you can help them with that part.
Be the BEST DAD you can be, that will be the best gift of all.
Skip the apology too, IMHO. Maybe sometime in the future it will be appropriate, but not now.
