If you have a motor mount going bad, are you suppose to be able to hear a noise of some kind?

I check a bad motor mount by lifting the hood and starting car, put the car in drive gear and hold brakes down. Give it gas enough to make the motor lean and lift up from a broken mount by the torque it gives. If the motor mount is bad, the engine will actually lift up about 2-4 inches on one side. To test the other motor mount, put car in reverse and hold brakes down and do the same thing. The engine will lift up from opposite side from a split or broken motor mount.
Remember when you try this, be in an open area so that you do not end up driving into something or someone when running this test. Hood up also so you or your help can see if the engine is lifting off the mounts completely.

Not usually. when a mount is bad, you can FELL the car **** when accell of a stop or changing from Drive to reverse. To check it, put on your parking brake, now hold the brake pedal with your left foot, now put the car in reverse, and press the gas slightly, the engine will move slightly, now put it in drive, the engine should shift slightly, if it moves a LOT like a few inches or moves upward, you may have a mount bad and it needs to be inspected.