“I don’t consider that a seriousl load”. Have you measured it? Screens do draw a lot of current, especially the backlight LEDs.
When I look at the National LM1117 supply characteristics it says max input voltage is 15v. http://www.national.com/mpf/LM/LM1117.html#Parametrics. So max input might depend on what manufacturer is used (ie just don’t assume it’s going to be capable of 20v)
Regulators convert one power supply to a different one. Power conversion generates heat. If you have a higher input voltage, at the same output current consumption, you generate more heat. Draw more current on the output, you need to draw a greater current on input, and you will generate (even) more heat. So if you’re trying to draw a higher current, and you are starting at the upper end of the voltage spectrum, then you’ll generate more heat. The on-board regulators are not designed for high heat dissipation, which is why all the suggestions are to use lower voltages when you’re trying to draw higher currents.
My suggestion for car powered Fez is to make your own step-down regulator to bring the ~14v (when running and the alternator is charging) to ~6.5v, and make sure it has a kick-ass heatsink on it so that it has a way to disspate the heat. Then feed the output of that into the regulator on the Fez. Another approach is to buy a car USB converter that outputs 5v and use a USB cable to power Fez, but make sure its not necessarily a “cheap” one but one that has a good regulator in it (hard to do without laying out the cash first I know ??? )