The ceiling is usually dead flat, flatter than a flat wall. That's the whole concept behind ceiling flat. Anything with a sheen will much easier show brush/roll marks unless done perfectly.
The color of the crown/base/trim/doors is of your choosing. The guys/girls at the paint store should be able to help you with these decisions, as it depends on your color scheme. The have color cards to assist you in deciding.
Eggshell or Satin or SG should all be OK for baths & kitchens, IF you are using a decent quality product. If you are only looking at price, all bets are off.
one more tip: Go to a real paint store, not Lowes or HD or Walmart, PLEASE!

I'd suggest posting this in the home decor thread...you'll get lots of suggestions there.

I would go with a flat ceiling, and keep the ceiling a the same color, but a lighter shade than the walls in the room. For example, with Ben Moore color preview, pick the second or third one down on the paint color strip for the wall and then one or 2 shades lighter for the ceiling. It helps unify the space and will help raise the ceiling. Trim should be painted all the same and in a white if your colors tend to be cool or a bone if your colors tend to be warm.