Hiya.
Firstly, as harsh as this may sound, you need to get a grip.
Secondly, you just seem to sound as though you still want a relationship with this man.
At the moment all you need to focus on is your baby, rather than him or his current girlfriend and what they're doing or what they're not doing with their lives.
I know you're probably scared and scared of being alone, but to me he does sound like he is a keen dad, even if he is a worm when it comes to relationships.
He has brought the baby things, he attends anti-natal with you, he wants to be there for you to support you during the birth and he's willing to come down every day when the baby is born for his 2 weeks of paternity leave. He is even coming down now to help you with bits that you can't really do now.
I will say this now though I think you wanting him to stay for 2 weeks is a bit much. If that was my partner I think that I would be exactly the same as Shivo and put my foot down. Even if it was him that begged you, at the time he didn't have another partner therefore didn't have to worry about these things, but now he has, he has to respect what she's saying concering this matter.
I admit yeah the nights are hard, but they only become difficult if you don't rest enough during the day, which I'm sure you will be able to do as he will be there helping you out, giving you plenty of time to get some rest. At the end of the day, you do need to get used to doing the nights alone, because he's not going to be there to help you at those times with you both being seperated.
I went through a similar situation when my daughter was born with my ex, and like you I didn't put him on the birth certificate but that does NOT mean that he doesn't have rights. You say he has threatened you with courts before and I can guarantee if you keep saying these things he will do it again, and actually go ahead with it. I got a big shock, once he goes through courts etc he will have plenty of rights. I just think you should be careful what you're doing.
I know it's not some perfect little family that most people want to have, but at the same time it is his baby as much as yours, and to be honest everyone is already slating him about being a dad just because it was a really bad relationship.
You just need to give your baby a chance rather than him of knowing her daddy and having her daddy be there for her without any trouble. The trouble going on is between you two and should not affect the baby or the relationship your little girl has with her dad.
You only need to focus on being a mum and dealing with the baby now rather than focussing on your ex and what he's doing with his life.
I really hope you've not taken this comment the wrong way, I am just talking from experience and I admit I was horrible at the time, and everything just turned around and bit me in the arse.
I think you both need to get a grip, and draw a line and be civil with one another for the sake of the baby because that's one thing you both have in common and have to prioritize above everything else, even the bitterness you have towards him.
Good luck with your pregnancy. :)