Yes, you can paint them. In the long-run I'd certainly replace them, but for the short term painting might work.

You need to clean them with a solvent, give them a light sanding with 400 grit or finer, then spray with a primer, then top coat using the color of your choice. Personally, I'd go with a hammertone finish--It seems to hold up better. I use Rustoleum hammertone because I can get it locally. The only drawback might be if the paint begins to wear off on your hands. I hope this helps.

Yeah, but you need to be realistic. Don't expect the finish to actually hold up when it's an item that people grab and twist several times a day.

Painting them would be OK. Sandpaper is hard to achieve total coverage. I would use a Scotch Brite pad, very easy to use. Clean with thinner or similar product. I would not use laquer paint...it dries fast but is very brittle and chips easily. A good primer and then the color of your choice. Should last along time

I primed and painted one door knob going to the garage and it came off .
Good luck!

You can try primer, two coats of color, and one coat of clear.