blue is perfect for oak. did you see all the colors I posted in my other comment? I put a few blues in there. this blue would look really nice w/the oak I also like the greens. Sage greens look great w/oak and will work w/your green frame. This is Silver Sage SW Svelte Sage Behr,,Quiet Moments bring home some other samples (if you use Behr, you can get the $2 samples. BM is at $7 each. )' these are some gray/greens and would look nice. I don't like the blue/grays w/the oak because those paints tend to run on the 'cool' side, whereas oak is warmer. Pale Oak (BM) may be nice for the other rooms. maybe try this link for the best wall colors w/oak cabinets https://www.hometalk.com/2389914/5-top-wall-colors-for-kitchens-with-oak-cabinets this one is Pleasant Valley

As with so many homes of this type, the problem isn't so much the wood but of the proportions of the trim. The baseboards, crown, and window and door casings are too skimpy. Also, the built-in bookcases around the door have no detailing--it looks like someone just installed plywood panels. Compare them to the photos I posted and you'll see the difference. Proportion is so important in designing anything, especially millwork. That being the case, and assuming that you aren't going to redo all the trim, you have 2 options. The first is to paint it all the same color as the walls so it visually disappears, and the second is to keep it stained natural and give a boost to the color palette--paint the walls a contrasting color. As Jan suggested, Hale Navy is great, also consider Narragansett Green, Newburyport Blue, or Weathersfield Moss.

I also suggest one color for walls and ceilings. You can leave the beam as is. The room will feel cozier (not smaller). For paint, a beige or cream. Gray would need a brown undertones. Here is a good article with color suggestions and photos. Paint can be hard, and needs to be seen in your space/light. I always like to get some of the sample boards at SW to paint and see during the day/night. Don't paint directly on walls so you can move around (and color bleeds through). Both BM and SW have "crap paint" (smaller cans, inexpensive paint - you would never use on walls) to buy for exploring.

I think you can keep the trim oak and paint cabinets a color if you want. I like that idea better than painting trim in some rooms and not others. My opinion. trim consistent. Cabinets whatever you choose.