Keep the wood ceiling as is, lighten the walls to work better with your furniture and accessories, especially with the glimpse of dark wood floor below. You can go with any light shade, not just gray or taupe. Perhaps you can get inspired by the "Franklin" sign in turq. See Houzz photos under "aqua" for the look.

Yes. Paint The Ceiling. First.

You can paint over wood paneling by properly priming it first. It looks like you have white ceiling elsewhere. Unless this room is completely separated, you might want to stick with that. Otherwise, a very very pale blue would be ideal.

Paint that first and then see how you feel about the wall color. You might love it with a new ceiling! If not, from the pictures we can see, the sky is the limit on color options. We'd have to see more to help you narrow it down!

I agree with sooz001, keep your wood ceiling as is - wood ceilings are very attractive. They give the room a very warm, cozy feeling and can also increase the value of your home.

Also, keep your white trim, but paint the walls of both rooms a BRIGHT, CHEERFUL color, such as certain shades of yellow, orange, or lime green. Try to find a shade that doesn't overpower the room, but bright enough so it's gives that "wow" factor, in a "good" way. Of course, take paint samples home first to see what goes best with the rest of your furnishings. Happy painting.