I'm guessing the rocket scenario is the first one and the asteroid one is the last. That being said...
1) The fuel in the rocket provides it with the energy it needs to propel itself into space. The rocket throttles at the bottom, allow it to break free from the earth's gravity by pushing against the earth's greater mass.
2) If you are inside of the bus, in a fixed spot not moving, and the bus is moving. That means you are moving in the same direction as the bus. If it stops, you're still moving at the speed that it was once originally moving at. Obviously, since you weigh less than the bus you are going to fly forwards.
3) There is no gravity or friction in space, so items won't slow down no matter what. Even if an object such as a meteor is flung into space with the tiniest bit of force it will keep flying through space at a consistent speed, since there's nothing to slow it down... unless it hits something else.
4) Looking back at the previous answer, the asteroid is flying through space at a consistent speed (which is really fast). Suddenly it comes in close range to the moon which is bigger in size, and thus has a stronger gravitational pull, and pulls the asteroid into it's field.
