First of all, I hope your DS is ok. His fall wasn't your fault. He has two parents and one of them was with him and should have been looking after him, despite their hangover.
Secondly, you are so lucky that this is your house in your name and you already pay at least 50% of everything and you're not married. You know what you need to do. This isn't a relationship or a partnership. No one should be treated or spoken to the way he has to you.
For context, my DH and I have an almost 6 month old. We both pay a proportion of the bills into a joint account and keep the same spends. As I'm on statutory maternity pay my contribution has reduced significantly and his has had to increase to make up the difference. No complaints from him and he asks each month if I need more from him to cover the bills.
He gets up for work at 6.30 and makes a bottle before he does anything. Whilst he gets ready I feed the baby. He goes off to work and comes home about 6pm - 7pm. Depending on if he's worked late the baby may already be in bed so he will always do the dream feed at 11pm in this instance so he can see her, albeit briefly. If he's home on time we tend to swap over each night so whoever does bed time, the other makes tea and will do the 11pm dream feed.
He will also get up in the night to see to her. We take turns because, although he works, he knows how exhausting it is looking after a baby and how touched out I am. Unfortunately she doesn't like to let me nap in the day. Sometimes on a Friday he'll suggest me sleeping in the spare room to get a full night rest and I'll reciprocate accordingly the next night.
They're currently downstairs watching Elmo right now and playing together whilst I get my arse out of bed.
Thing is, he didn't want a child but compromised because he wanted me and that was my deal breaker. He's stepped up far beyond what I ever dreamed he would.
My point is that he's a partner AND a dad and is acting accordingly. Yeah we're BOTH knackered and we're BOTH skint but we're in it together. He doesn't turn his phone off after work and disappear until 4.30am because he needs a break because he knows I need one too and he's not that selfish or disrespectful.
You're already looking after your beautiful baby single handedly and providing everything he needs. Please don't stay in this situation because both you and baby deserve so much more. Please give yourself a break if you can and if you have any friends or relatives who you trust that can give you a couple of hours to do something for yourself then please do it.
This selfish man child isn't going to change and you'll bend further and further and end up breaking to make him happy. Please get out now whilst you're able to.