"Should I suggest H share D16's care?" YES, he is her Dad. He is responsible for raising her to. Who cares if it restricts his coming and goings. He being a Dad is a responsibility, IMO, that cannot be ignored.
"A couple days after he left he told me he thought he would call me up sometime and ask me for a date. Should I remind him or just wait and see what happens?????"
Personally I would let him bring it up. He mentioned it to you, which means that he is thinking of you. Keep up the PMA he is thinking about you and expressing that to you. That is not happening in my sitch.
This is a long road, sometimes the pavement is smooth and other times you cannot move an inch without hitting a pothole. But in the end with or without our S the road will lead us to a place where we are much stronger, confident, and better people.