I recently hired a plumber to fix a few leaks and drips. He spend about
45 minutes trying to fix my leaking bathroom sink, but failed and said I
am SOL. He couldn't get the seat out because it was stripped, and
smoothing it out didn't do the trick. The drip is getting worse daily.

First question I have is should he have charged me for 45 minutes of labor
even though he failed to fix the sink?

I'm also now stuck facing a costly remodel of my bathroom. The sink is 50
years old and the vanity is custom. If I replace the sink, I will have
to tear out the vanity, the ceramic tile splash-back, and put in new
flooring because there is nothing else on the market that fits. Not
something I want to do. The plumber said it is possible to pull the sink
out of the wall so that the faucet valves can be replaced, but this would
likely be more expensive than installing a new vanity sink.
Sigh...

Any advice? Should I call a 2nd plumber? I question the man's competence...