I don't like how he wants to call the shots, he wont agree to anything I say etc. Shes 5 months old, he should prove himself to me that he can bond with her.
I just don't know how to deal with it all.
Honestly it's quite obvious that you don't know how to deal with it.
You're angry and indignant. You wish he would just disappear and you're stomping your feet and holding your ground because he won't go away and give up. You feel like you have no control in the situation so you're making rules and dragging your feet in an effort to assert control or make him give up - and it's obvious it's not working because he's not letting it, that makes him a good dad (and remember, he IS a dad).
You are being unreasonable and you know it.
Your boyfriend has your car and he takes it to work, therefore YOU HAVE a car to take the child to the contact centre to facilitate contact. You have a child that needs to see her father. Either your boyfriend gets his own car, or he organises his own transportation on visitation days. You need to be more flexible about this and the court will think so as well. You're putting your boyfriend before your child.
It's time you stopped trying to ignore everything her father wants and instead try to turn what he wants into something you can handle, this is called compromise. You don't need to do everything at once, but you should at least try. You need (and should want) to be able to tell your daughter that you tried. You need to be able to look her in the eye when she asks about her daddy and is crying about her daddy not knowing her and know that you did everything you could to facilitate a relationship. That's what a good parent does.
Her father also doesn't need to prove ANYTHING to you about how he bonds with HIS daughter. You don't get to decide that. If you believe she's at risk, then you'd have something to say. Don't just make up things to make him look bad. Be the bigger person here.
I think most of all you need to get counselling. You sound very angry with yourself for being in this situation. Sounds like you harbor a lot of guilt about having a child with some guy you dated for less than 2 months, and someone you don't even like.
So call the contact center nearest you and set up a time, or go there to check it out. Make this a priority. Tell your boyfriend to find his own way to work for a day, or drive him in and pick him up after wards. Then organise a schedule, via email, with her father to make the visits. Leave him to bond with her, don't sit there and hover. Watch from a distance if you can. Over time this will start to feel better but you need to start to do it first.