OK - answers to your questions:
1. The carb does not have an accelerator pump. It has only the main pump. (In other words, it looks like the one in your picture except there is no pump on the passenger side, only the driver side.) I replaced the pump diaphragm (and the power valve also) during the rebuild.
2. A cold start when the car has been sitting for several hours requires one firm press of the gas pedal to the floor, then one pedal pump while cranking, and it fires right up. When starting from cold, it will run at idle for about a minute, smooth at first, then gettting rougher and rougher until it dies. Once it's warm, it will start right up, but idle is very rough and it dies in about 10 sec. Engine runs fine as long as I give it some gas.
3. I replaced the PCV valve last night - no change.
4. I'm not sure how we would use a vacuum gauge since I can't get the engine to run long enough to use it - however my friend has one and he is bringing it over tonight - any suggestions? I can try the carb cleaner trick also, but since I can't get it to run at idle, I'm not sure if this will work either.
5. Timing - I really don't think it is timing, because we cannot hear any misfiring, missing, or other strange sounds when the throttle is opened up. I would think that a timing problem severe enough to cause engine stall would be more audible. Also keep in mid that this problem happened suddenly one day; the engine ran fine before that.
Tonight we are going to try running the engine from a standalone gas can (directly into the fuel pump) to rule out any fuel contamination issues.
I also saw in another thread on here today that someone else rebuilt an Autolite with the Napa kit, and had to adjust the float about 3/16". Now they have the same problem I do. He thought the spec in the kit's instructions might be wrong, and the bowl is now filling too much. So I'll try moving the float down about 3/16" tonight.
If none of that works, well, I borrowed a timing light yesterday, I'll try checking timing. Any other thoughts?
EDIT: By the way, have I told you guys thanks for all the help and patience with stupid questions, y'all are the BEST!!