His family dynamics are obviously not what you’re used to, but you married him, so unless you want to cause some serious upset and ruin his family relationships I think you’re going to have to suck it up! He does favours for his family, be happy that he cares enough.
Unfortunately when you have children they are your responsibility, so no one should have to offer to look after the baby while you go to the pub. You chose to have a baby, You look after it. The only exception to that is your husband. Who should have had the common decency to alternate with you and let you go while he stayed with the baby. You could have told him that after the first night he went out, and hopefully the situation would have been sorted.
I’m sorry you feel worried enough in your own home to not want to stay there alone, perhaps you should look into moving. And if he went to look after the dogs for a weekend and left you alone I don’t see what the difference is him going for one night.
Why should MIL pay to put cousin in a hotel when she has family who all help to support and look after her? When she knows she can call on her son for a favour as he is reliable? She probably knows that she spent years of her life raising a considerate son who will help her out when she’s older.
I’m sorry but this post does make you sound quite selfish, maybe have a little breather, re-read and try and see it from everyone else’s point of view. I would be pretty proud if my child grew up to be someone family could rely on. It’s up to your husband to decide if its all too much and taking the piss, if he doesn’t think that then you’re probably going to have to suck it up or risk alienating yourself from the family.