1) if you look at those datasheets for the devices, many of them will go down to 2-ish volt or even 1-ish volt for newer devices. So they don't have to work at 5v. The arduino happens to be designed to work with 5v. But if you were to jump that regulator, you can go down much lower than 5v. BTW, they don't need that 5v to be precise either.
the vreg chip gets hot using anything except the 6v adapter.
With the 6v adapter the vreg ran cool, but with any of the others which input 11-13v, it got warm to hot.
with any of the others which input 11-13v, it got warm to hot.
X 6 digits = 60mA.
A typical backlight whether EL or LED will draw 100 mA.