D.B.
F. - I think it's clear that his actions are all about controlling and manipulating you, inconveniencing you, and watching you jump through hoops. If he wants to see his daughter, he can show up at your door with 48 hours notice minimum, with a car seat and a diaper bag and a stroller. You should have your coat on like you are heading out - hand him the baby and LEAVE!
If he doesn't show in a half hour, take her out and leave a note on the door that you waited and then you had a date or an appointment and could not wait for him. Do not be there when he arrives.
Do not meet him at the mall, do not meet him anywhere unless he wants to meet at the police station where it is warm and safe for her. Stop doing these thing for him and he will eventually stop asking.
Get a lawyer or mediator to put a visitation schedule in place if you really want it, and then start the proceedings to get child support. He needs to stop yanking your chain, and the only way for that to happen is if you get strong and stay strong. Your daughter is too little to know what's going on, so get these structures in place before she is aware enough to be disappointed.