What's happening:
I noticed a few days ago that the first floor A/C
system did not seem to be cooling, but I didn't have
time then to check it out. When I went back to
investigate yesterday, I found that the unit blower
would be on but the air wasn't coming out cold.

The blower is a separate motor from the compressor (the outside unit).

I noticed an unusual dimming of the living room
lights when the A/C unit came on -- not just a
temporary dip-then-back-up, but a noticeable drop
to maybe 2/3 brightness that didn't come back up
again.

Stalled compressor. Might just need a capacitor.

I checked and all circuit breakers seemed to be
set -- but I flipped them off and reset them to
'on' anyway. Back in the house, when I turned
the A/C unit back on, the same bank of lights
that dimmed before with A/C operation (living
room et al.) now wouldn't come on at all! They
did eventually come back on, but it was after I
went outside to reset the single-throw breakers,
and I believe it took several minutes even after
I turned the A/C off (these are standard
incandescent bulbs, not fluorescent or halogen).

When you manually tripped the breakers, one or more failed to close. Your
second trip fixed that. Several minutes???????????????

My guess is your only problem is the compressor.