Help with alternator replacement 2000 Ford Mustang GT?

Hi, I'm fairly new to this forum. My battery and alternator recently failed on me so I decided to take it upon myself to purchase and install a new set. It was a fairly easy job and I was done in under 20 minutes. Now, when I bought the new alternator it's pulley had a diameter that was about 1/3 of an inch smaller than the previous one in my car. I didn't put too much thought into it due to the fact that the sales guy assured me it was the factory size. I took it out for a spin to make sure everything was good, and right off the bat I noticed the battery gauge drastically dropping. I thought maybe the alternator just needed a second to get going, so I kept driving. That s when my problems started. I could feel the car starting to die before it started shaking and sputtering pretty bad. I was close enough to home that I wanted to try and make it to my driveway. The car completely died a few times while driving, so I engaged the clutch and threw it in 2nd gear and popped the clutch. I had a lot going on while finishing up the alternator job and stupidly forgot to reconnect the two cables to the alternator. It looked as though the belt was what was smoking, although I'm not entirely sure. Can running the alternator with the belt attached, neither of the two cables though, ruin the new alternator? Can anyone explain what happened? Thanks so much in advance!

Oh and my car is a 2000 Mustang GT.

Sounds like that pulley on the new alternator is too small to use with the same size belt. You likely saw smoke from the belt because it was slipping on a pulley. Either get a smaller belt or a larger pulley.
To verify that is the problem, check the belt for slick, shiny looking marks - indications that the belt was indeed slipping.

The alternator was not damaged. You really should charge the battery up before starting the car. Charging a dead battery. Now that can hurt an alternator. I would not worry about the pulley. 1/3 of an inch is only 1/6 inch radius. Plus the alt will go faster not slower. You did the job too fast. What was your rush. Hasn't this approach bit you in the azs many times before?

First of all you should have gotten an alternator with the same size pulley as the old one.
Yes you can fry an alternator if it's not connected properly. That can put it into uncontrolled maximum output. Go to Advance Auto and get the correct alternator. The belt was smoking either because you couldn't tighten it enough because of the smaller pulley or you seized up the alternator.

Find the charging system fuse. It is likely blown out. Replace it with a 15 amp fuse instead of the stock 5 amp. I bet your old alternator was fine. This happens to Fords all the time.

I call the dealer with the Vin # and get the amps of the alt, and then tell the parts guy. 1/3 inch is small, but I'm ocd. Get the right part. Smokes not good lol. But if it works now, role with it.