I agree that Termini Station is not the nicest area for a single female to stay. I doubt anything would happen to you but you probably would not feel comfortable walking around in this area at night. There are a lot of sketchy people hanging around.

As for places to stay try Nicolas Inn, Fellini Inn or Gullivers Lodge.

I am not aware of any museums that are only open in the morning during the winter. Most museums are closed on Mondays. The Vatican Museums are open on Mondays but closed on Sundays.

With 4-5 days as a single traveler you will have ample time to tour Rome.