Couple of things standout here as not making sense. Why did it overheat or more precisely, why do you think it over heated?
A P0171 code is not for the transmission. It is bank 1 too lean. What it is saying is that based on inputs from the front oxygen sensor, the PCM (computer) had to add at least 20% more fuel to get the right air/fuel ratio. The most common causes for this is a dirty MAF sensor or a vacuum leak.
If your engine is running too cold, you will get e check engine light and code for that. Even 2 degrees low will trip it. The top of the radiator is not a good indicator.
You don’t need to drive it 100 miles, but you do need to drive it under all the conditions needed to run each monitor. Usually drive till warmed up, then 10-20 minutes at 45 mph or higher, pretty steady, should do it. Have all your paper work and receipts from the garages that have worked on your vehicle so if you get pulled over, you may get a “by me” on this if the cop is in a good mood and you don’t disrespect him/her.
But you have to address that misfire code. Have you changed plugs recently? Why did you go straight to the coil packs?