As for the heat, does the gauge go up? The heat behind the axle could just be the exhaust, it may just sit very close to the underside of the car. I would assume that if it were something in the rear axle generating that much heat you'd probably have heard some horrible noises by now, but you may want to check the rear end fluid just to be safe.
Otherwise if the gauge is actually rising and the car is nearly overheating that could be various things. The most common problem would be the thermostat, but obviously make sure you check the antifreeze level first. Other than that it could be the radiator is clogged, the fan isn't kicking in on time or at all, or possibly a head gasket.
Good luck with the new ride!