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: http://www.cdc.gov/niosh/npptl/topics/respirators/factsheets/respfact.html.
NO. Need special mask. Here is a good reference: http://www.nbcgasmask.com/RadProtec/index.htm.
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.