In F1 they sometimes do the same thing, but not only for tire heat. As long as you're driving, the air flowing through the engine cooling system provides some cooling. If your opponent is standing still, the engine cooling isn't optimal and could lead to overheating. I don't know if this is also the case for the bikes though, maybe their cooling is more than adequate.
No, the bikes will overheat too. There was a race a year or 2 ago, I don't remember what series, where several bikes were having problems getting back to the grid, or the light tree was having problems starting, and the waiting racers starting freaking out, and finally just took off and did a 2nd "warm up" lap to cool their bikes down.
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