Always check tire pressures cold first thing in the morning before driving (or less than a mile if you don't have a way to do it at home. The vehicle placard pressure is a starting point but needs to be adjusted (usually upwards for on road cars) depending upon conditions. In the winter, if your car is garaged, you need to make an adjustment for the lower outside temperature. In summer you need to make an adjustment to compensate for the higher temperatures.
All tire pressure recommendations are based on a few assumptions, which may or may not equate with your driving conditions:
1. Temperature is 18C (65F). At 38C (100F) add 4 psi. For vehicle placard pressure of 310 kpa (45 psi)
2. Speed is faster than 105 km/h (65 mph) add 3 psi (Tesla most likely already does this).
3. Load is what the engineers thought the car would be at most of the time--this may be lower than the GVW. (Add a bit when traveling with the family on vacation.)
Note that this is very different from tire pressures for track use, but then tires used for track aren't expected to last very long.
> How much does the pressure increase just from driving?
The correct answer is: it depends.
What it depends on is the starting inflation pressure, tire size, and
vehicle weight.
The actual process is that the tire flexes and creates heat. This raises
the tire pressure, reducing heat creation. This cycle of flexing--pressure
increase--reduced flexing continues until the amount of heat generated is
equal to the amount of heat lost. At this point (called thermal
equilibrium) the tires will not gain any additional pressure.
What you are shooting for is no more than a 10% to 15% pressure increase
from driving. There is also a pressure increase from a rise in ambient
temperature (hot sun, black tires, this can be 10% or so).
If your tires are 25 to 50% low (unfortunately very common), you can
easily get way over 25% pressure increase (you will be able to fry eggs on
the tires, worse the compounds will start to deteriorate).
It's normal to put additional pressure in the tires in hot climates.
Getting this right can, in some cases, double the life of your tires.