I really love this stuff and can't find any fault with it. Your style is incredibly cute, but different from typical cute styles that we used to see in games of this kind. Your style has its own personality and it's good. The rag doll look of your characters is especially awesome. For the record, I prefer the originals -- before the thicker outlines.
Instead of a thicker black outline, if you really feel like something needs to be done for readability, you might take a cue from real designers who worked with that hardware. There were many games back then, especially in the later years of Gameboy's lifespan, that used a "white" (lightest shade of cabbage green) outline, tracing the "black" outline. Does that make sense? I'm sure I could dig up an example if you need it, there were plenty.
The only problem is you'd either have to reduce the size of your sprites by 1 pixel on each side to account for the extra outline, or just throw the sprite size limitations out the window. You said you didn't think you were exact with that already, though, so it may not matter. Anyway, that would be my suggestion, if you think you need something to improve on perfection.
Otherwise I say congratulations. This is the kind of game I would have played for graphics alone. That's a real accomplishment when you're working with 4 shades of cabbage.