I have used epoxy putty for similar repairs. Usually found in paint departments of hardware stores.

Photos would help.
Depends how big the chips are.
If white the oil based paint would have yellowed so matching could be difficult hence the suggestion to repaint the whole door or…

Forgot to ask. What roller sleeve are you using?

That's a high possibility. Give the rougher area a good sand first.
Never a good idea to paint ceilings with heaters on or even air conditioners in the sumner.
Hope all goes…

I used to do paint inspections for a living but this has me stumped a bit.
Does the surface feel rougher than the other areas?

When you say chalky, does chalk come off on your hand when you rub it?