Mother in law trouble
If you see her, tell her that, for religious reasons, you are giving up anything connected to gossip, including talking behind people's backs. Tell her you want to hear how she is doing, from her. (You can't rely on your boyfriend's sister-in-law to tell you accurately what your boyfriend's mom said about you, in any case. Right? Isn't this SIL mad at you anyway?)
If your boyfriend's mom starts in trying to make peace between you and the in-laws, tell her that's very sweet of her, but you really don't want to discuss other people behind their backs, and hope to work this out eventually. You can keep saying "work this out eventually" for a loooooong time. In the meantime, you can have a relationship with the mom, herself, and cut all the drama.
If you aren't living with this boyfriend, and that persists, you might not have to deal with these complicated relationships in his family.
(I know, it's probably not part of your religious beliefs to avoid gossip, and maybe you're an atheist. When you say, "for religious reasons," they don't have to be YOUR religious reasons.)
Most people don't realize how much allowing people to tell you what other people are doing can wreck the relationship. If you have to have a relationship with your children's grandmother, the way to stop being furious with her is to insist on hearing about her from her and about other people from those other people.
The best way to screw up a family relationship is to allow one person to sit in the middle and control all the threads. It's bad for your relationship with your partner's sister-in-law and it's bad for your relationship with your partner, which sounds kind of strained, but it's also bad for your relationship with the mother-in-law. (Or mother-out-law, not sure what you call the mother of the father of your children who you aren't living with.) She will be nicer to you if you insist on her telling you her own story. This is my been-there-done-that advice.
Ask her about what your partner was like at your kids' ages. Tell her your kids like Legos (or whatever they like) and ask if her children liked them too. Make the smallest of small talk and find ways to get the conversation on topics where you don't feel triggered. Do what you can to honor her positive impulses and redirect her. (That's why I said to tell her she was kind or sweet for wanting you to make up with the other relatives.)
Ignore anything she says, or is reported to have said, to anyone else. It might be true that she's acting like a blue meanie behind your back, but you have to insist on failing to acknowledge that. (If she says these things to your face, of course, all bets are off--but I'm guessing she won't.)
A few questions:
Where are you getting any income to cover your rent, food, medical care? I am guessing your boyfriend is not paying any child support (buying diapers and wipes is NOT supporting his children)?
What does your boyfriend say about all this? Does he tell her it's none of her business and that she needs to keep her mouth closed as regards your life AND that of your boyfriend's siblings? If not, then maybe you need to realize that his family's (Mom & siblings) opinions and needs come before those of you and his children.
Your boyfriend's brother and sister-in law, are they married? What did they say or do that was so awful?
Look at it from your boyfriend's mother's view: you are living with your Mom and (I'm guessing, as you don't refer to this) not working. Her son is living with her boyfriend. Since you are not living with her son, she doesn't see the both of you in any type of committed relationship, especially after 4 years and not living together anymore. Despite having children together, you live apart from one another, with your "parents", not independently, as a couple. He comes over a couple of days a week and helps out. You're 25, in her eyes, you should be living on your own, together, and supporting yourselves (and the children) 100%. She doesn't see two adults, she sees two children, with children, living off others.
She's not your mother-in-law. You're not married, she's your boyfriend's Mom, that's all.
If it causes so much grief for you, tell her to STOP (and, make sure your boyfriend does, as well). If she continues, tell her you won't allow her to see the grandchildren anymore, and stand firm on this. But, realize, in doing this, you may be cutting her from your life, and that of the children, forever (be ready to explain this to them when they are older and ask about their grandparents). You will also be alienating your boyfriend from his family, as he'll need to make his choice of you or them (could you abide his decision?). Know, too, there is a very good possibility that she will drag your name through the mud. If you can live with that, be done with her.