I'd do a pale gray on the walls, and leave the moulding white - nice and neutral, and not too busy, but still with a little pop. (I'd do the crown, base, and windows white too, if it were allowed.)

Any pale neutral on the walls, leaving the molding white, or vice versa. But PLEASE do not paint the natural woodwork!

I agree with the first comment. Gray on the walls and white trim will do the trick! Good luck on your painting project.

I have a similar setup in my living room. What I did, and one idea you may like to consider, is using chalkboard paint on the interior wall moldings, leaving the "frame" white, and then paint the rest of the wall a color of your choosing. It really brings out the wall design and with the chalkboards it allows you to further add your own personal touches to the room.

Taupe or light taupe walls would offset the white molding nicely and warm up the room
more than a grey but still easy to paint over if/when you move out later on.

(Behr paint - Vintage Taupe 750A-3 or the Feather Grey 750A-2)

My apt walls are similar. I haven't done my living room yet, however my bedroom is gray (Behr Amazon Stone)with white (Behr Cascade White) trim. Whatever color you choose, make sure that the white is really white. It makes everything pop. For my living room I was thinking some kind of a neutral color, the white trim but maybe some removable wall paper within the frames. Not sure, but these walls can be a lot of fun.

Oh and don't paint the wood!