Apple TV on additional tv

You can connect an additional Apple TV to the TV in the bedroom, sure.

You can also just move the Apple TV between different TVs as required.

You can't send a signal form the Apple TV in the living room to a TV it isn't connected to if that is what you are asking.

Yes. I have 3 AppleTV's in 3 rooms. Obviously you need to buy another AppleTV for each room. You can set them on all sync so they are all identical w/ your apps etc.