Not sure if insets should be caulked( maybe they are supposed to move?) but prime first, then caulk.

if your doors are frame and panel construction the panel must be allowed to float freely in the frame. It will expand and contract with seasonal humidity changes much more than the frame. You probably want to paint in the dry season when the panel is contracted so that it doesn't then shrink further and show an unpainted stripe. If you seal the panel to the frame the wood *will* move and something will give--you can hope for the paint or caulk bond, but it's also possible that the panel will split. When making new doors it's much better to finish the panel before installing it in the frame, but that's not usually an option in refinishing. Just try to not let too much paint gather in the rabbet that holds the panel.

Even if the panels are meant to float freely, I caulk them anyway because it really looks like hell if you don't. Usually, it holds up. Just use a good caulk like DAP Dynaflex 230.

We use GF milk paint which is almost as thin as stain and presents no problem with build up in the crevices. One downside with is that it is extra work to clear coat cabinets after painting them with thin paint like that.
As far as the side panels, if paint does not hold on them, I would consider covering them with edge banded plywood first.
Whatever paint you use, if you do not spray it I would thin it out as much as possible and use a foam roller to avoid brush strokes.