Check Ikea they had some that were not deep. I made my own sink base at 16” and put a 12” round vessel sink on top but it’s a powder room so it’s enough for in there.

Is this the exact spot where the mirror will be hung? I'm only asking because looks like your current vanity is on the opposite wall and I want to make sure that the space you'll have for the sconces will be the same.

How wide should the vanity be if the wall is 50 inches wide? Not sure if it should like a piece of furniture or built in

You can put a 1x2 or 2x2 furring strip behind the vanity to seal that gap and maybe put some cove molding over it to make it look a little nicer on the corner. The furring strip might work better if you have a backsplash and it can be at the top of the backsplash (but level with it). It would add a mini shelf of sorts in the gap. Cove molding would reduce the size of what you could put on the mini shelf, but you can play around with images to visualize it. If your designer has skethcup or CAD they should be able to do a sketch for you. The side drops don't really bother me as I've seen a lot of vanities like that. Leaves room for magazine rack, trash bin, etc. I will be installing a vanity that will have to be bumped forward a little to allow room for the P-trap and I will be putting a little shelf level with the top of the backsplash behind it to catch stuff that falls.