I missed the original post but ...
Your right about the undercover on our Supra's. Don't leave that off or the cooling system will be effected.
That electric fan is controlled by the A/C high side pressure. The switch is near the passenger tire. I forget were the relay (maybe passenger kick) is.
Your Supra cooling system should not need modifications. Mine has no problems at all crossing the desert mid day when its like +110f degree. For sure 100 mph for hours would not be any problem.
A bad engine driven fan clutch (fluid coupling) can cause bizarre problems
** If you have coolant disappearing from the overflow bottle and bubbles that is bad news. Especially if it has been hot or had coolant leaks.If you see milky looking stuff on the inside of your oil fill cap. BHG
Hope that helps. Dan
BTW: Head gasket kit for mine was +$350