Battery charging issue, need advice

The car is my girlfriend's 1998 Toyota Tercel.

So here's the symptoms:

• After driving for about 20 minutes, the car battery light came on.
• The headlights seemed dimmer than usual.
• The radio kept "rebooting" (probably from low voltage).
• Once we got to our destination the car wouldn't start again.

Now, normally I'd say "oh, the battery must be dead," but the next morning we were able to start the car and drive it to Sears where they tested the battery and said it looked okay. They also said it's probably an issue with the charging system.

Another possibly-related symptom: the car has been emitting a high-pitched chirp for a while now, probably due to a loose belt or something.

I don't know much about cars, but my guess is the Sears guy is right, it's something in the charging system and not the battery itself. Replacing the alternator would be expensive so I want to avoid that unless it's necessary. So, my questions are:

• How could the battery be "dead" the night before and then somehow work the next morning? That one upsets the thermodynamicist in me.
• Could a belt issue cause this kind of behavior? We'll definitely be fixing the belt regardless, but it would be good to know in advance if this could fix both problems.
• Does this sound like an alternator problem?
• If it does turn out to be the alternator, how feasible is it to replace it yourself?

Thanks for the help.