I know they use a Manned Manuevering Unit (MMU), but if they do won't space station or shuttle just leave them behind.
They don't use the MMU anymore. It was decided that it's too risky and they can get the same stuff done with tethered spacewalks and by attaching the astronaut to a manipulator arm. Astronauts nowadays do have an emergency jet pack called SAFER, though there's never been an emergency where it's been needed.
Next time you're on a plane, or in a car, or on a train, I want you to take an object, toss it in the air, and then catch it. Then imagine what it would have looked like from an outside observer's point of view. The astronauts are moving at the same speed as the space station or shuttle (neither is accelerating or decelerating during spacewalks). Using the MMU just allows them to accelerate and decelerate around the space station or shuttle, thus changing their relative positions.
No. The station isn't actually being propelled by anything, it just has momentum and nothing to slow it down. The astronauts, having been propelled to the same speed as the space station, carry the same momentum. So they wouldn't get left behind unless there was some kind of force that slowed them down but somehow had no effect on the space station.
The MMU has control thrusters on it that can be used to keep the astronaut at a neutral relative velocity to the Space Station. Meaning that the two will not separate to far from one another.
