On hot days, after a few short trips, the car won't start at all. The red battery light comes on and nothing happens until it cools down for at least an hour. The A/C hasn't worked in a couple of years. Could these be rrelated? Thanks.

Sounds like the Alternator going bad, it is heating up under load and cooling down causing it to work again.. But my guess is in the next few days it will go.Also the battery is building it self back up a little as well. But if you keep this up the battery life will start going down.. Take it to auto parts place they will check the alt for you... FREE Autozone ,Advanced Auto and some NAPA's will check it for free. You want even have to pull it out..The AC could cause it if the clutch locked up. But you would hear a loud squall from the belt. So my guess is the Alt is just going..

Well, if you had that battery since 2001 I would also have it checked... The auto parts store can also check the battery to see if it's holding a load. Extreme heat and cold are a cars batterys worst enemy.
You could check if the alternator is not charging properly yourself with a voltmeter connected to the battery leads and while having a helper cranking you can monitor the batterys output.
At standby the volts should be at 12-12.5V. while cranking It should not sink below 9.5V (I believe) and should come back up to around 13.5-14V at fast idle.

Strauss (similar to AutoZone) were doing a $15 "electrical system check" recently, which checked the battery and the alternator. They might have done other stuff as well, but those were the main things. Of course, free is even better than $15, so...