The builder did put some sealant where the wall meets the floor (looks
a lot like caulk to me, actually, except that it's grey instead of
white...).

There is a creek running maybe 200 feet behind the house, but that is
at a lower elevation even than our back yard.

As for the foundation drain - the builder called it a foundation drain
and hasn't mentioned anything about draintiles, but I'm not a
construction expert and am not even sure what is a drain tile.

The basement is the same size as the first floor of the house, which I
guess mean's it's around 1000 or 1100 square feet. We don't normally
run the AC except maybe an hour in the evening.

There are water pipes in the basement, but I did check those first
thing and there wasn't any obvious problem with them.