My question is this - if I choose to file for a work permit and the
inspector comes out to check the new work I've done, would he also check
any of the old work and fail the inspection?

If he has no reason to inspect the old stuff, he probably won't even notice.
Have you given him a reason to poke at the old wiring?

In particular I found that the receptacles in the kitchen are not
grounded, but have 3-pin receptacles (non-GFCI).

This is illegal so he can shut you off as soon as he discovers this. Your
wiring may be pre-code, but as soon as someone put the three prong outlets
in without grounds, your wiring became illegal.

There are probably other issues too, but anyway, would any of these be
part of the inspection or is that "grand-fathered" work that is exempt?

He can flag anything that is unsafe, illegal, etc.