Symptoms of bad alternator

What are some of the symptoms of a bad alternator? Our 1992 Corolla won't go over 20 - 30 MPH and runs real bad.

A terminal voltage across the battery lower than 13.5 Volts while running, low headlights, While idling, the car will stall if the negetive teminal is disconnected.

Sounds like a fuel delivery problem [filter, pump, corroded fuel line], or plugged Cat Converter

We have fixed a lot of things (exhaust manifold, timing belt, oil seal, etc,) but now the car takes forever to get to cruising speed and once at cruising speed, it runs well. Do you think we might have an idling problem or a plugged cat converter? The cat is less than 3-4 years old.

Is the check engine light on? if so, you can do a self diagnostic check on it, or take it up to the nearest auto zone, where they will run a free diagnostic check to see what codes it's putting out. If you are mechanically inclined, I have a haynes manual which says how to get the read out at home with a jumper wire, and what the codes mean.

Not getting any check engine codes.

A quick check for it, is to run it up to a local parts store...Autozone, Advance Auto, or Oreilly's...all of which will do a free battery/alternator check for you. However this problem doesn't sound like an alternator issue to me. If it's running real rough....i.e. shaking and rumbling, have you checked the plugs and wires? that can cause power loss to the engine, and a very very rough idle.