You'll want 3/8th's inch glass. 1/4" is so thin (do they even sell that thickness?) and will look cheap.

I have a Fairmont vanity. It's nice enough - not cheaply made like a "builder's supply" vanity. The drawers do fully extend and are soft close (and they work well).

1/4" glass is flimsy, 1/2" glass is great, so 3/8" is good.

Ask your shower door people. With a sliming door that is framed I would think 1/4" would fine. Maybe Millworkman will chime in.