The nails on my pressed-board wood siding on the house I bought a few
years ago have become countersunk.
Maybe they weren't nailed into the studs. Maybe they started
countersunk a little bit and moisture has done its job. Maybe the
siding is lousy. Whatever.
What's the best putty/spackle/caulk to use (after the pressure washing
but before the painting)?
Anything else I should know/do/consider?

Sand down to bare wood, then apply wood putty, leaving it a little
thick so you can sand it down level after it dries. Put a coat of
primer on all the bare spots before you paint the whole thing.

Before you do a lot a prep work, check the siding to to see if you'd be
better of replacing it. The fact that the nails are showing up as
depressions means the wood around them has swelled.

--
Steve Bell
New Life Home Improvement
Arlington, TX USA