You are right that the Moon is slowly moving away from the Earth. Satellites in orbit around the Earth also require nudging to keep them in the proper orbits, as you point out. There is no one single effect, though, but *several* which account for this behavior.
If the Earth and the Moon were both points instead of rotating spheres, and otherwise alone in space, then they would orbit each other with only the slightest changes to the orbit, over immense time. They would actually move closer, but in a profoundly small way -- practically unnoticeable over the current age of the Universe -- because of a small loss of energy to "gravitational radiation".
But the Earth and Moon are indeed both spheres, and in the case of the Earth, it's figure is rotating far faster (once per day) than the Earth-Moon revolution (once per month).
The Moon's gravitational force, or actually the *difference* in the Moon's gravitational force between the Earth's near and far sides, raise tides on the Earth. These tides lag slightly the rotation of the Earth because of friction, and the result is that the Moon sees a slight tug from these delayed tides that is not strictly on the line connecting the centers of both the Earth and the Moon.
As a result, two things happen: the Earth slows its rotation, and the Moon gets a nudge that pushes it into a higher orbit. So indeed, the Moon is moving further away from the Earth -- and the Earth is slowing down. Eventually, the Earth/Moon system would become "locked", just as the Moon is now locked such that it always has one face turned towards the Earth. This would take longer than the life of our Sun, so it's a moot point.
The important thing to remember is that the "energy" of the entire Earth-Moon system stays constant -- the energy of the Earth's rotation is being transferred to the energy of the Earth-Moon orbit.
In the case of satellites, there are various effects at work. Near to the Earth, there are bits of tenuous atmosphere that act frictionally on satellites and slowly rob them of orbital energy such that they drift closer and closer to the Earth, eventually to reenter the dense atmosphere in fiery fashion. Higher up, a slight "lumpiness" to the Earth's gravitational field (after all, we have continents, oceans, lots of things with slightly different density -- along with an equator which is stretched out relative to the poles) slowly moves satellites in their orbits -- so these orbits require correction if the satellite is to be be in a precise place at a certain time.
Much further out (communications satellites 22,000 miles out in "geosynchronous" orbit) , satellites drift primarily because the Moon and Sun tug on them. They must have thrusters for "stationkeeping" so that your rooftop antenna can continue to point to them in the same spot in the sky over months.
As to what happens when a satellite runs out of fuel -- for all except for satellites at great distances from the Earth, their fate will be to burn up in the Earth's atmosphere. Satellites which are orbiting the Earth at distances a significant fraction of the Moon's distance may have several fates -- they may enter the Earth's atmosphere, they may have orbits disturbed such that they may eventually hit the Moon, or they may be tugged out of Earth's orbit into orbit around the Sun. It's almost like a celestial game of billiards. When several significant gravitating bodies are involved, motions can get very complicated! But near any particular body, things behave generally as if that's the only body at work, because its gravity is so much stronger than that of the rest. These effects are generally slow ones. Satellites make only tiny corrections with tiny thrusting jets. It's all that's required.