try disconnecting the sub from the amp, and then check to see if the amp turns on and stays on.
also, you have got a good start trouble shooting, but are looking at the terminals wrong-the 12 volt in, and remote line should have no difference(though a slight difference is ok) in the power readings-- both should be recieving at least 12 volts(+/-3 volts), the only difference is that the remote line is a car controlled line to turn the amp on and off. the 12 volt line should be a constantly powered line directly off the battery.
the other issue you could have is that the amp is connected to a faulty ground. try either moving the ground, or cleaning the area/bolt that the ground cable is connected to. to test this, measure the voltage difference between the ground post on the amp, and different ground points near the amp-- there should not be any more than +1-2 volt deviation from any point(this would be placing the +lead on the ground on the amp, and testing different points with the -lead.)