She sounds just like my MIL was when I had DD1. I remember feeling like you do now, I was so tired, living away from my family, not a clue what I was doing, and DP wasn'tmuch better.
On one occasion when DD was a few days old MIL arrived on the doorstep and told me to go to bed, she'd sit with DD for an hour. I was so wrecked I just said screw it, I'll go. I was awoken not long after to the sound of her sister leaving my house. I hadn't invited her in, MIl hadn't asked if she could call, and never mentioned that she was there. When I asked her she just said oh, she was passing and wanted to see the baby seeing as you haven't really brought her over to our house yet. It was a good indicator of things to come.
Now, nearly 5 years later, and one and a half more babies, I've finally learned how to deal with her. We've also moved far away from the ILs.
I think I started to crack when I had DD2. DPs brother and his girlfriend came over when DD2 was a few weeks old to tell us they were pregnant. They were only there a few mins when MIL arrived to tell us the news herself. Anyway, we were chatting and congratulating and DPs SIL was saying how she's nervous about having the baby and being at home etc alone. I said its hard at the start but its fine, and other encouraging sounds. MIL then remarked I don't know what you're talking about Flowermum, I was here every day looking after you when you had DD1!. I just stood up and left the room, couldn't stop the tears. We barely saw her for the first few weeks unless we went to their house adn the odd time she called to let me sleep for an hour. I think everyone got the message, and DP finally manned up and got rid of them. And then I had a breakdown, crying, screaming, the lot.
She also took a pop at my mother one night when she said she thought my mother was a disgrace for not coming to the hospital the morning I was rushed into emergency surgery because my appendix burst. She came the morning after, when I had come to instead. I nearly hit her, really, it was very bad. I got emotional and everything, which I never do in front of people. I made a half arsed defence that my mother lived at the other side of the country, had a business to run and four other children to look after and anyway, I was unconscious so what use would she have been anyway. Cow.
Basically, what I'm saying is some MILs are awful, and I don't know why. Until I had DD1 we used to get on ok. God knows what happened, the surname thing did insult her. I insisted on double barrel. And also point blank refused to name DD Wallace, her maiden name. My surname is Wall. HELLO!
Anyway, the way I deal with her now is I NEVER ask her for anything. I've very rarely asked her to help out with DDs re childminding, from day 1 I'vebeen as independent as I could.
I know lots of people say you're lucky to have her to help with your baby, but hard and all as it is to have a small baby, distancing yourself from her might just be easier. And tell your BF to MAN UP!