First of all, I'm so sorry you are going through this. Your primary focus should be on your and your baby's wellbeing. surround yourself with people who love you.
In terms of the birth, he has no right to be there. It is an intense experience, do not be pressured into anything as its vital you feel safe and supported.
For the travel system, honestly, if you can, buy your own. Hes using it for control dont give him the satisfaction. I bought a fancy all bells and whistles one but then needed a smaller one for travelling so got it cheap off facebook marketplace and its just as good. I use them both.
In terms of next steps, it does depend. I am generally of the opinion a child has a right to know their father unless it would be unsafe. You dont have to answer but unless you honestly fear that he would hurt the child or use PR to continue abuse you, the birth certificate is a legal document so he should be added.
In terms of contact, it is in the best interest if the child to coparent. At the baby stage, little and often is best. Only a few hours but regularly. How close do you live to eachother?
If you have a trusted friend of relative could they be the point of contact between the two of you for a while to let emotions calm and keep your final weeks as smooth as possible?