When you start the car it uses an electrical curcuit to run the ignition. When you release the key that curcuit changes to the one for driving. When you let go the key and it dies that means that part of the curcuit is not energised. It could be in the ignition switch or possibly somewhere else in the curcuit.
The reason your dash lights brightened when you reved the motor with the starter engaged is because you used the starter as a generator and it increased the electrical power in your system. This is VERY bad for the starter and not good for the electrical system of your car either.
Has nothing to do with the heat. It is an intermittant short and most likely in the ignition switch.
As to the fans if you have A/C then with it on at least one fan should run. If you do not have A/C then the fan should only come on when the engine is over a preset temp. What does "incredibly hot" mean? Was the guage up near 3/4 or more? if so the fan should have been on. If you have a heating problem I would get the fans checket by a mechanic. If not I would not worry about it as they are probably working and you just don't notice it.