First, apart from welcoming you here I'd like to say it is pretty often the case you feel both love, pity and anger at someone. I'm not in the least surprised you feel this way too. Your sister's actions have lead to this situation and the kids are basically stranded.
They are lucky though in your mum and you loving them, not all have that.
You said before al this started you had ongoing depression and BPD, and these will of course be made worse now.
To cope wiht hte situation both oyu and oyur mum need to be in the best condition possible, and also have support. May I asked if you have a medical team, perhaps a GP and a psychologist to help you , give you therapy and maybe meds, and importantly if you trust them a sense of perspective.
You cannot shoulder all this by yourself and it is important you know it. It takes more than one.
You mentioned going home. Do you live away from your mum, father and the kids? If so is there anyone in your life to give you support? Perhaps a partner, another family member or a freind. It makes a world of difference to have that. When I was very ill my wife had her mum, both practical and emotional support, it got her through.
Your father sounds truly horrible and I'm wondering what sort of state your mum is in. By rights she ought to talk with the people that are used to this, 1800RESPECT - 1800 737 732, would be a very good place to try. They can see things are not straightforward and do not offer simplistic answers. Come to think of it it might not hurt for you to ask their advice.
Kids needing you may not be all bad, if they can return love in time that is pretty special.
Look. I know I've not talked about all the problems, but first things first, and that is to get you the best you can be
I hope we talk some more