They are likely covered with some sort of polyurethane to protect the stain. You would need to sand down/off to apply more stain. With a bit of scuffing, they would take paint.

Sliding doors look fine. No. Definitely would not paint the sliding doors. Anything painted will, inevitably, need to be repainted. Multiple layers of paint will eventually alter the ability of the door to slide and/or seal properly. Add drape panels with a valance instead.

They were stained by the previous owners and several areas are blotchy. I don’t think there’s any poly on top. When I wipe with a wet cloth, some stain comes off into the rag.

Before painting the trim - add fabric panels and cover the sides of the trim. Hang panels on each side of the doors/windows. Put curtain rod a few inches from the ceiling, with fabric running to the floor and your ceilings will look even higher. Fabric does not have to be wide enough to cover the doors or windows if you have other coverings, but the added fabric with the hardwood will reduce the echo factor, soften the room and add a nice texture to compliment the paint color. Looks like you could even run the curtain rod across the entire length from the right of the door all the way to the corner. Fill the corner with panels, and carry the rod across the door running perpendicular. Not my pic, but hope it conveys the idea

Thank you. This is an idea I’ve considered too.