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Not really. Once something is in orbit, it's just going to naturally stay in orbit unless acted on by an outside force. That's just basic inertia.

There is some amount of atmospheric drag in low earth orbit which will eventually cause satellites at that altitude to lose speed and fall back to earth, but that drag is small enough that the satellites should be able to remain in orbit for years without needing any additional thrust.

The way it works is that the satellite has very high velocity tangent to the earth- so they are constantly falling to earth, but they are moving tangent at the same rate that they are falling- because the earth is a sphere, the ground "falls away" from underneath them. They're constantly "missing" the Earth.

The moon doesn’t need any propulsion to stay in orbit around the Earth. A satellite is just a very small and very close moon. The fact that it’s closer than the moon just means that it will orbit faster than the moon does.