In my opinion it's important for a child to have a relationship with its father. My dh is an absolutely fantastic father, but his ex has really destroyed his relationship with his 2 kids with her. Funnily enough he sounds a lot like your ex. He has a criminal record, but he had it when I met him and he had it when I chose to have children with him. It's from his past and really not relevant to his ability to be a good and loving father. And I suspect your ex is the same...otherwise you wouldn't have chosen to have a child with him.
Dh is always late for everything. So is my father. And so is my boss. All of them are wonderful responsible fathers. It hasn't stopped my father from becoming a famous scientist and my boss from becoming a successful architect. It's just bloody annoying. I have learned to plan around it, since it doesn't seem likely to change. Certainly not a reason, in my opinion to wreck healthy father daughter relationship.
It's completely unacceptable for your dh to call you "names" in front of your dd. I think that if you broach this subject without bringing up any of the others, you may come across as wanting to correct one thing and not just nagging about lots and therefore have more success at changing the behavior. My dh used to do the same because that's all he ever heard in his family growing up. Now he really does try to modify his behavior and only say positive things about and to me in front of the kids. Of course, since he is male, he does need frequent reminding.
Not fastening the reins on the buggy is dumb, but not really life threatening. I'm not quite sure how old your dd is, but you could put her "in charge" of reminding your ex to fasten the reins. I did this with my dc's and it actually worked really well. Dd would screech incessantly until dh fastened her up. She did it with the car seat too.... Ds was less on top of the reminding.... but then again he too is male.
Up the slide thing is really not something to be worrying about in the big picture of things. If you teach her good manners and playground etiquette, she will take that with her through life regardless of what her dad lets her do. Save your energy for the bigger problems.
Don't really understand the forgetting to contact him issue. Doesn't he have your phone number? If he doesn't, then he should and if he does, then just ignore the sulking....like you would that of a little child.
Men are really dumb about things like being straight forward about dp. Just ignore it. Dh used to do stupid things like not tell his ex that I had cooked dinner for the kids or something similar. He had some theory that it would get her riled up. Of course what got her riled up was the lack of the full truth (and the fact that's she's completely bonkers...but that's another issue).
Altogether, I think that you are really doing your daughter a big favor by making the extra effort to keep her father in her life, despite his dumb and annoying male behavior. My friend once gave me some advice for dealing with men...after everything they do and say, add the words in your mind "mental age of 12." It really does make it all a bit more understandable. And good luck....it really will be worth it. I have a friend whose son has just turned 18 and he has become a really wonderful and well adjusted young man. She worked very hard at making sure he had a relationship with his father even though the father was a lousy husband and it involved big sacrifices on her part. It really paid off.