Can I paint directly over finished doorframes floor moldings?

You want to clean them thoroughly and maybe give them a rough sand paper sanding and wipe again. Sanding helps the new paint to adhere better and longer to the wood.

Lightly sand to remove any gloss and roughen the surface for paint. Prime with a good stain blocking primer. Then paint.

Yes, as long as you clean and use the same type of paint or oil. If you don't know what type of paint, you will need to sand, prime and paint. Good luck.

I've found a good washing down with TSP (in most grocery stores) gets rid of the dust and grime and hand oils....usually no rinsing needed. It's most likely the previous paint (if you're not sure about it) is a latex - so go with a semi-gloss or glossy (your choice) and use lots of painters "Blue" tape to tape off the walls....it will save you extra work touching up the walls after. You also will want to tape under the baseboard to protect floors, whether it's carpet or whatever......then take your time...happy painting..