Families come in all shapes and sizes. You are still a family, just not the one you thought you were going to be. That isn't a bad thing if being together was driving you both crazy with constant arguments and upset. At least this way your dd can hopefully have 2 happier parents, who can parent her better apart than they could together.
Time is a great healer. But you've had hardly any time for that to happen yet. These things take many months, not days or weeks. Keep pushing for the help you need from your GP and midwife. If they can't give you pills ask what help they can give you. One day you will look back at this time and realise just how strong you were/are. But for now be kind to yourself and allow yourself to grieve for the relationship that is now gone.
You don't have to send your child to an unknown address. It is perfectly reasonable for you to request and be given the address your dd will be staying at. You wouldn't send her off on a school trip or to a friends, or even for a day out with grandparents without knowing where she was going to be now would you...! If it went to court the judge would be unlikely to think it an unredsonable request for you to know where your child was staying. Keep that in mind and ask him would he be happy if you moved and refused to give him your new address so he didn't know where dd was while with you?
I really don't think any responsible parent would allow their 2 year old child to go off with no clue where they are. Especially if that child was potentially going to be staying away overnight. What if he didn't return her and wasn't contactable? Where would you tell the police to start looking?
Is he insisting you leave your own house while he is there? What's to stop him having the new woman, or anyone else, there without your permission.
It is your home and you shouldn't be made to feel uncomfortable to be there.
He's not doing you a favour. He's spending time with his dd and you are helping to facilitate that.
If the only way to do that is by him coming to your house, and you are happy with that, then he can't expect you to go out for hours on end. Not least with you being pregnant and needing to take it easy!
So he can't have it all his own way and expect to suddenly go from having dd for a few hours in your house to having her overnight the whole weekend.
And you can't have the help you want without some compromise either. So you need to try and agree contact at his house, but he needs to let you have the address.
Unfortunately you will need to find someone else to help with bath/bedtime, or let him have her overnight if you can't cope every night and need the break.
You might find you can cope with it better if dd has spent an afternoon with your ex before he brings her home anyhow. Is he having the contact at your house at weekends or in the week? How is he fitting it around work?
IMO your dd is a little young to be staying away the whole weekend. At this age contact should be short and frequent building up to an overnight, and eventually every other weekend when she is comfortable and happy with that. Don't be pressurised into doing what he wants. If your dd isn't ready for it then he needs to go at her pace.
Every other weekend by school age is more reasonable. She's only 2 and going through a lot already. She will already be more clingy and regress when the baby arrives. So you want to have a good routine, and agreed contact, in place by then if at all possible. Don't say no outright to overnights, but explain that it is not yet. Once contact at his house is going well you can move to an overnight once a month, then every third week, then every fortnight. Once dd is comfortable with that then move to two overnights. Slow and steady and your dd will get there. So if that's what he wants then he needs to work with you for dd's benefit over this.
Stand up for your dd and yourself. Don't let him bully you into thinking it has to be his way or no way. if he really care about dd and the baby he will put their needs ahead of his own wants and needs. Don't be afraid to remind him of that.
Keep strong. You have a lot to deal with in the coming weeks and months. Take one day at a time and you'll get there.