Repairing damaged marble veneer in kitchen

Yes, you must remove all the paint.

All the weight of the mortar and stone will be depending not on the mortar bond to the substrate, but on the bond between the thin layer of paint and the substrate. That bond, which was designed to hold only the paint itself, is sure to fail in time.

I have repaired two different situations like this:

• thin brick pavers were mortared to painted concrete, which resulted in the pavers (and mortar and paint below that) releasing from the concrete starting one year later.
• thin stones were mortared to painted stucco, with failure timing and mechanism similar to the previous scenario.

In both cases, grit-blasting would have been the appropriate (and ultimately cheaper) treatment.

That is not the "door frame" but rather door trim and specifically it is crown molding. It will be nailed or screwed. Get up on a ladder and examine it from the top. Take pictures of what you see.

You will probably have to remove this element of the trim and reinstall it, but don't be in a hurry to tackle this. You could cause a lot of damage if you don't do it properly. Do it properly and you may not even have much (or any) touch up painting. Do it wrong, and you will make a mess that will require a professional to restore and will cost a pretty penny.

Honestly, a competent trim carpenter should be brought in to fix this. It appears another element has been pulled from the wall, and there is a trim piece missing, or paint missing, from the end of the horizontal trim above the door. It looks like the door trim was not complete. Is this a new house?