Do you usually AA items/characters that can be against multiple backgrounds?
Yes and no. You would AA the interior of the sprite and could try some "selout" which a lot of people don't seem to like or maybe some light affected outlines that get lighter near the lightsource.
I was trying to imitate a style choice by using the thick lines and blocks of color on the pyramids.
If you want to do the thick outline approach similar to that of Yoshi's Island or whatever it might be best to give everything that outline (at least all the background and level elements) to keep more of a consistent style.
I'm not totally sure what parts need better buffering. I've never used the terms "color ramps" and "buffering" (I really need to extend my vocabulary) Are you saying that I need to have some kind of small gradient where I have shadows to make the transition smoother?
Basically the way I see it, any large color plane/block needs some sort of AA between it and the next color except in the circumstance that the colors are very close together and would not benefit from AA (or a buffer shade).
I made a little example of "color ramp" and "buffer shade" for you. In simplicity your color ramp would be, for example, all your greens from light to dark (or dark to light). Buffer shade would just be the shade that is in between two other shades.