Late to the party. Exactly right about Nitrogen, it still follows the gas laws and expands, it is the moisture it does without. same reason it is used in airplanes, so the very cold temperatures does not freeze the moisture.
To check tyre temperatures, you really need a tyre probe as the IR devices are just reading surface temp and that is affected by literally everything the tyre rolls over. Drive through someone's puddle where they have washed their wheels or the damp patch under the tree etc. and it all cools the surface. You are looking for no more than a 20 degree spread across the tyre, if you have just come round a long right hand corner before entering the pits, then the outer edges of the left tyres should quite naturally be hotter that the inside and so on. Understand also that the outside edges of tyres will naturally, no matter how good your camber settings, will show the most wear. If they don.t then you have too much camber.
Manufactures will show increased tyre pressures for load or fast driving because the tyre walls flex during rotation. That flex causes causes head in the sidewall just as bending a coat hanger back and fourth does. While you are not going to exceed the elastic limit of the sidewall, like you do in the steel, that heat, if excessive, can and does cause degradation of the rubber. Anyone that has driven any distance on a deflating tyre will know just how hot and ultimately smoky they can get. Rubber does not like high temps and nasty shit happens. You have all seen unmaintained truck tyres in the central reservation of the motorway.
Again, the "book" message of not letting air out of a hot tyre is nothing what so ever to do with track driving, it is to do with setting cold pressures when hot. The cold pressures for my van are about 57psi. When loaded, on a motorway and a dry, warm day, they can easily rise to 72 and above. If I were to then set them back to 57, I would be down around 42 when cold and not in a good place. In track driving, when you expect to be putting heat into the tyre because of the loads you are putting through it then, firstly the only true reading you will get is hot pressure and cold pressures are just an attempt to get to those hot pressures. It is also exactly what manufactures do with their cold "door post pressures" they know the tyre is going to heat in use, it is just their very technical guess at a mid range pressure, suitable for all situations they expect to encounter on the road, including the heating effect of the sun. On track you know you the pressures are going to change, even if it is wet. The trick is (irony alert) don't believe the crap you read in forums. I have lost count of the number of people that say "my car understeers" to find out they have listened to the "internet" and gone out at 20 psi on their 1.5 tonne fwd monster. And secondly, warm your tyres up gently before smashing them over the curbs. It is highly unlikely on a "typical" car that you need less than 25 cold, unless you are in some featherweight device. The reason you see all slick shod cars weaving like crazy on their green flag laps is because they do expect to generate lots of heat and even rwd Porsche's and the like will be going out with pressures as low as 18psi knowing that is going to come up to 30 or so. You will even hear commentators saying "it will take 2 laps (in the race) for the tyres to be to temperature".