Hi and welcome to the board. I am so sorry to hear about the loss of your father. On top of that this situation with your sister is just making things harder I imagine.
Prior to her recent behavior had you ever thought she had BPD? It sounds like you were quite close at one time.
You talked about what she wants and asked for but what do you want in the relationship right now? Do you want to maintain contact at this point or do you need to take a break? How would you like your relationship to look in the future?
She has a history of cutting and suicidal thoughts. Is it better for me to leave her alone at this point or keep trying? Is it better to let her continue to make demands within reason? Or am I setting us up for future problems by not asking for more trust and cooperation?
I would say it is better to leave things as they are for now. Give her a chance to self soothe and return to her baseline. It sounds like a lot of this was triggered by your father's death and that is not uncommon. People grieve differently and death can bring all sorts of dysfunctional behaviors to the fore front. I am not making excuses, rather I am trying to look at the context. Am I getting it right?