Going down the captain pedant conversation path here, but technically all satellites also need to burn energy to stay in orbit or will eventually fall. The only ones who don’t have achieved escape velocity
Some Lagrange points are stable and therefore will not decay toward the Earth outside of other factors. (Because these systems are never sufficiently isolated, in the eternal view, therefore, also still require energy, though much, much less.) Though, of course, an object at a Lagrange point may still not technically be a satellite of earth. https://solarsystem.nasa.gov/faq/88/what-are-lagrange-points (though by the NASA definition above I'd argue that they are)
To further add some (maybe helpful) pedantry, the boundary between an airplane and a satellite is usually taken to be the point at which the velocity required to remain aloft via aerodynamic lift exceeds the orbital velocity if there were no atmosphere.
No, they don’t. In the absence of drag, which only the lowest satellites have, they just stay up there forever. The fuel is needed for orbit changes and correcting drift due to gravitational instabilities.
IANAP ("I am not a physicist"), but any two objects in orbit around their common center of gravity are slowly radiating energy into space in the form of gravity waves. This is why LIGO reports its chirps. Of course, this isn't very much energy, but given enough time all should orbits collapse.
I am a physicist. The gravitational energy loss from planet + satellite scale orbiting bodies is so small as to be orders of magnitude less than, say, the influence of gravitational anomalies like the Himalayas, or the tidal pull of the Moon.
Are any of them truly free of drag? Like, are there 0 molecules of atmosphere at some height, or just entirely negligible amounts of atmosphere for all practical purposes?