... It also doens't expand like oxygen does. ...
Umm, all gases expand exactly the same way. The equation is PV=nRT, also called the ideal gas law.
If the absolute temperature goes up 10%, then either the pressure or volume (actually, the product of the two) must also rise 10% -- it doesn't matter whether the gas is nitrogen, oxygen, helium, or even steam.
There are exceptions to this rule at extreme limits, when gases are said to be "non-ideal," but that does not apply to air versus nitrogen in tires.
Note that any liquid component will change the dynamic. So if you have entrained water in the air supply, then, yes, as the tire heats, the water will vaporize into steam, and that will increase the pressure (or volume) by a greater percentage. This is one reason why the air supply should be perfectly dry for airing tires.
... On air, I get 10psi rise in tire air pressure on the track in three laps. With nitrogen, only 4 psi! ...
Then you had water in your air supply. See above.
Nitrogen molecules are larger than O2, so they do not leak out of tires as quickly.
OK, this part is true -- but just barely. The difference in size is minute. Also, if this claimed benefit is really operating at any significant level, then, theoretically, your tires would be mostly full of nitrogen after a few top-ups anyway. Air is 80% nitrogen to begin with, so after first airing up the tire, it will be 80% nitrogen. Now let's say that all of the oxygen leaks out, and you find your tire pressure to be 20% low. You fill the tire back up with air -- now you have 96% nitrogen and 4% oxygen. If that 4% also leaks out, when you top up again, you'll have 99.2% nitrogen, and only .8% oxygen. Etc.
Also, there is zero humidity in nitrogen, a huge advantage to tire life. Tires do not like humidity inside them!
Well, OK, but you can also have 0% humidity in "air" too. Note that nitrogen is made by processing air, not from some other magic process. That processing removes the oxygen as well as other trace gasses, and, of course, the moisture. But it's not perfect -- some trace gasses and even some water vapor remains. It is certainly easier and less expensive, though, to just remove the water than to remove everything else, too.
Uhh, yes, exactly the same way that nitrogen does. Again, see "ideal gas law" above. Steam is a gas, which is not to be confused with liquid water. Liquid water, as I wrote above, can be entrained in the air supply and can vaporize inside the tire, increasing the pressure. Perhaps this is what you are thinking of.
... Oxygen is needed for corrosion of both metals and rubber. ...
Buswarrior is correct. This is really the only significant advantage to using nitrogen over (100% dry) air -- oxygen, well, "oxidizes." Oxidation of the rubber ultimately weakens it.
That said, the likelihood that oxidation on the inside of the tire can have a measurable impact on the life of a heavy-duty tire is very small. First off, the tire is exposed to oxygen for its entire life up to the point where it is mounted and filled, and so there is already an oxidized layer. That layer actually inhibits most further oxidation. As the tire flexes, there will be some additional oxidation. But most tires will wear out long before this oxidation can become a factor in casing failure.
I personally believe that fleet programs seeing a measurable difference using nitrogen fills are actually reaping the benefits of having a consistent fill source. By mandating nitrogen fill, you are precluding the driver from filling his tires at some crappy truck island that might be loaded with moisture.
The single largest cause of tire failure is under-inflation. Take the money you would spend on nitrogen fills and invest in a good tire gauge, and use it before every trip -- you will be ahead of the game. And you'll compensate for any of those sneaky oxygen molecules that permeated through the casing.
FWIW, JMO, YMMV, etc. etc.