I don't think it is a good idea to shut the car off in the staging lanes unless there is a long delay such as when someone drops oil or something like that. My car's temp is always higher when I restart after shutting it off unless it is off for a longer period of time.
If the OP was getting no wheelspin at launch, I see no reason to lower tire pressure. That would just increase rolling resistance. No wheelspin at all tells me that he could launch a bit harder. Then if there is significant spinning, I could see lowering the pressure. My best times have always been with a just bit of spinning to prevent bogging but not enough to hurt the time.