The heat discharged into the air will warm it up some, but it will probally still be rather cool, so all the heat won't go back into the radiator, most will still be in the air. It would be better not to run the trans cooler in front of the radiator, and for it to have it's own fan.
I don't like those zip-tie like strapps that go through the radiator, they peirced my old radiator and I have seen them rub thier way through fins on ohter cars as well.
Transmissions do not have any components that need heat to function correctly. Trans fluid properties make it where the viscosity stays very consistant, however it does change some due to temprature. In the classes I have taken, and all the trans manuals I have, as well as in my experience overcooling is never a cause of trans failure (unless of course the trans fluid actually froze) and the only side effect of a cold transmission is shift quality, it may shift a little firmer with a cold transmission. However, overheated transmission fluid is probally the leading cause of transmission failure. Heat destroys the transmission fluid's properties. 220*F is as hoter than you want to get, and over 250*F the fuild is burnt. At enging operating temp, 180-195*, your getting kind of hot, and after a pass throught he converter, it's borderline. Durring a race, it gets real hot. Powerbrake (stall check, WOT with the brake 0n) your car for 10 seconds, then put your hand on a metal transmission line before the cooler, hot ain't it?