Is a supersaturated solution still a solution?
You seem to be getting it wrong. A solution with crystallized solids at the bottom is not the same as supersaturated solution, and has never been called such. It is called "solution with precipitate", "heterogenous mixture of solution and something", "water with mud", or anything in the same vein.

A supersaturated solution is a solution, not just technically, but in all sincerity. It is homogenous, clear, transparent, and you'll never tell it from ordinary solution - I mean, until you touch it, at which point a precipitate would start to form.

As for your second question: yes, there are quite a few compounds that would dissolve in water incredibly well. ZnCl2 comes to mind... Wait, it's simpler than that - common table sugar (sucrose) would do! It dissolves up to 2000 g/L, how'd you like that?