Questions:
- Should I use pressure treated 2-by-X or is it OK to just use primed
and painted ordinary stock?

i would use pressure treated. another option would be cedar.


- If I do use pressure treated stock (ACQ probably), can I paint it
and if so what special preparation (if any) is required? (one guy at
the local borg store said I need to wait 3 months for the wood to
dry before painting)

pressure treated can be painted just like any wood. it is true that if
it is soaking wet (as it often is) it need to dry. that would be the
advantage of using cedar...it usually comes dry.

- What do I use to secure the new 2-by-X stool to the foundation? I
would prefer not to have to nail or drill into the old rock
foundation. Is there a glue, caulk, or foam that will do the trick
of both sealing and securing the stool to an obviously irregular
stone masonry surface? (I could add a little mortar to cover any
gross imperfections)

you could probably use a construction adhesive like liquid nails or
similar.