Driver Ed Events (track) - you will be fine, no problem.
If you have the track skills already - then you'd probably be starting with the Z51 anyway as it has some of the first common mods people do when they really get into it. Not that it is required, just that like with anything there are always more toys to buy, ;-), should you wish.
silvervette79, welcome aboard. Like Ronin2014 stated, anybody can it the track 1/4 mile or road course. I've did the 1/4 or 1/8 mile drag strip, but I've never driven on a road course. I would have to ride with somebody who has road track experience. I think that the auto would do better at the strip, and the manual would tear up the road course. There are some road course racers who are members, that could answer your question better than I could. If I ever get a chance, I would take my C7 on a road course!
Very doubtful to overheat the trans on an autocross course. Most are laid out to stay in a lower gear and are in the neighborhood of one minute long. Since you are not going to be running 20 to 30 runs back to back you're not going to get close to overheating. Plenty of bone stock automatics (sedans, minivans, full size trucks, etc) run and I've never seen any issues.