OK, this is an update, and a good lesson learned. I took everybody's advice, and bought a new OEM alternator. I spent about 3 hours getting the old one out, and the new one in, only to get the same results - ABS lights and no tranny control/lights. On a whim, I started checking fuses throughout the Rodeo, and found the tail light fuse blown. I replaced this 10-Amp fuse, and everything is fine.
To confirm, I took the time to put the original alternator back in, and it also works fine. Rodeo is running great.
So, lesson learned: Check the fuses first!! Now I'm going to try to find a wiring diagram, and find out what that tail light fuse has to do with the control circuits for the ABS and tranny. Amazing.........
Now I have a "slightly used" alternator to sell on E-bay.
Thanks again for the advice. It was a fun project for my 14-year old and I. Regards.