Any mechanics correct me where I´m wrong. But since none has answered yet, I´ll try:
Lehpron: Are they filled with regular air or some cold gas to keep them from heating up when their used?
The initial temperature of the gas isn´t relevant as it adapts to the environment´s temperature anyway.
As far as I know, some airlines are using nitrogen to fill their tyres because it has better long-term properties and is also better on temperature changes. (I don´t remember the details very well, here.)
Other than that, conditioned air would be used, as far as I know.
Lehpron: I've seen pix of bizjets that don't have main bogie doors, so do they burst in the pressure difference when the plane climbs?
No. The gear well doors are not airtight. They are only there for aerodynamic purposes.
Even if the plane would fly to outer space, the pressure difference would only be one athmosphere while the normal operating pressure of the tyres is several times that. (I might be wrong here, but I believe even the gear wells of the space shuttle are not pressurized.)
Lehpron: When measuring pound-per-square-inch, how much of the tire is used for calculating?
, so I can´t tell for sure...
But the pressure within the tyre is distributed evenly over the entire inner surface. So the pressure would be measured through the valve as it´s done for a car.