What can be done about a vanity top not flush against the base?
It would not be uncommon to install blocking around on the inside top edge of the vanity cabinet to butt up against the top assembly to hold it level without the top rocking around. Such blocking could be pieces of wood 3/4" thick that you glue or screw to the cabinet sides at the necessary height.

As far as that edge along the sides....only you the installer is apt to be looking up under there to see that. I would think that you can just leave that area open. For sure I would avoid the temptation to smear it full if caulk or sealer.

Another thing. You probably want to get the sink top all installed without general regard to where the drain down spout is located relative to the drain piping. The P-trap and the drain piping is all designed to be adjustable to fit to the sink installation and not the other way around.