If you AA the outline to pure white, it'll look weird on darker backgrounds, so i'd suggest not doing so.
I agree
When you set your sprite over a dark BG, it'll have a weird outline...
You can paint the BGs no problem, I think some games have done that... It's not going to look bad unless there's noise or the colors blend.
Just try it out and fix the things that look bad as you go, but it's not going to look bad just because of the clash of different "mediums".
As for the field sprites, you just have to "tilt the base" away from the viewer.
The typical top-down view has the characters tilted slightly downwards (you can see the top of their heads and the top of their shoes).
You don't look at them from the side (as you have them right now) nor do you look at them from the top (where you would
only see their head and some shoes sticking out)
It's kind of like a 3/4 perspective, except here it's vertical tilt and not horizontal