I'm an animal behaviourist the cat is stressed, upset and miserable. That amount of shedding is usually synonymous with stress or the wrong diet. He is checking up fur balls probably because he is over grooming due to the stress. I wouldn't expect even a long haired cat to be throwing up hair balls that often.
The poor thing has been heaved from everything it knows and dumped in a hostile and negative environment.
I know none of this is your fault, you never asked for the cat.
The fact your bf found it acceptable to punt his,responsibilities over to you without so much as a discussion is a very good indicator of his lack of maturity. Expecting others to clean up his messes.

I won't repeat what's ready been made apparent by the others, but there are so many warning signs here I would proceed with caution or better still go back to living separately until he can behave like an adult.
I wish you all the best and hope you manage to find your way. Take care.