Will a remote key operate the doors if another remote key is in the car?

No the other key will not operate the door when another remote key is already in the car.Same key has to be used to operate the doors.

Edit(razor512):

If a car comes with 2 remotes for the alarm and the door locks, then both can work at the same time, even if one of them is inside of the car.

The area where things get tricky is if the remote for 1 car is working on another car. In which case, the factory used the same code on more than 1 car, or someone used a radio monitor able to listen at the frequency that the remote runs on. The signal sent out by the remote is then recorded and replayed in order to spoof the remote. (this is why it is not the best choice to get remote starters, someone monitoring the frequency range can pickup the door unlock signal as well as the remote start signal, then replay it in order to do malicious things to the car (though this should be rare since most criminals are idiots)