Do hospital face-masks protect you from inhaling radiation if there's radioactive stuff in the air?

Surgical masks No. Surgical masks won't cut it (pun). You did ask about Niosh Masks on a former HT. See:
NO. Need special mask. Here is a good reference:
Depends. Most radiation is carried in particulate material which might be blocked by most face masks. It would be rare that it would be such fine/small material that it would pass through the mask.