Hello, I would like to suggest you to opt for a light shade as creamish brown or pale yellow as it would suit the dark brown dining table very well. Apart from that, you can opt for olive green as you want to, but I don't think it would be a great combination with dark brown. Before opting for anything, try the samples at backyard to have a look on what would it be look like in future. Best of luck and do upload photographs here after painting.

Well, white is a neutral and works with anything so no matter what color you choose it will look good with the wainscot. That said, I think you're real question should be what color can you use with the brass fixtures and yellow tile. Yellow goes well with a lot of colors but I believe beige is not one of them. A lot of the beige tones tend to lean towards the pinks and it can be very difficult to find the right tone. Red looks great with yellow but I think a bright red room w/yellow tiles may be too strong. In my opinion the red would work better if used sparingly with towels or your accessories. A color you didn't mention that is popular and I think would work well with the yellow, white and maybe pops of red is a light silvery gray.

Hi--- Benjamin Moore's White Dove or Sherwin Williams' Creamy. The room, furniture, and rug are dark, the wall will look great painted a very light cream. If you prefer a darker hue, I suggest you consider changing the rug to something lighter to add contrast in value. Hope this is helpful for you. Good Luck!

Deeper color would be cool (not that leaving it white is wrong). Hard to comment on what exactly the color should be..can't see your backsplash very well. Would you describe it as grayish blue(that's what I see on my monitor), or..? why is it trouble that you still like your vintage posters? I think there's no trouble at all..makes all sense when one's attached to the art he owns, otherwise why to have it. I think the poster will work very nice in your kitchen, whether you choose a color or leave it white.