Thanks Snapperfreak, I got it all back together so I'll include a pic of it here. I've seen your pics of a snapper that you completely restored and repainted and mine isn't nearly that cool. I did spend a lot of time cleaning the engine today, but all I had was some engine cleaner, screwdriver, and towels to get in the grooves. I wish I lived close to my Dad because he has a parts washer and a sandblaster at his transmission shop.
I wanted to ask if you could take a look at the throttle assembly above the carb to see if I hooked it up correctly? It wouldn't fire up when it was back together and I'm thinking that the second butterfly valve on the carb isn't opening on the engine side because it's hooked up wrong. I was too excited to work on it and regret not taking pictures as I went. I took two pics of the throttle assembly, one is on full choke, and the other is on stop.
I'll mention that I replaced the ignition coil today and I am quite a novice at this kind of thing. I feel very accomplished getting it all back together without any bolts left over. I feel good about how I installed the new coil, but I wouldn't write it off as the problem.