[blykins]

If the engine is running, then the torque converter is turning, which means the front pump is turning.

As for removing the driveshaft while towing.....if the rear wheels are spinning and the driveshaft is turning, then the output shaft is turning, which causes the speedometer cable to turn, which could cause the odometer to rack up miles.

I wouldn't worry about letting the engine cool....I'd let the engine help on the braking. It's not like you're constantly driving down a hill for the life of the car anyway.