I guess it depends on what you define "regular" as. If the satellite is low enough, atmospheric drag will slow it down and decay the orbit, so there will need to be periodic boosts to correct the orbit (like with the ISS). For Geo satellites, atmospheric drag is a negligible contributor, but other factors such as irregularities in the distribution and strength of Earth's gravity, solar wind, and Earth/Moon/Sun gravitational interaction will cause disruptions to the orbit.
Geo satellites must perform North/South and East/West station-keeping burns to stay in the proper orbit. In addition, regular burns are conducted to desaturate reaction wheels. There are different techniques (high vs low thrust) and different schedules to achieve all of the required burns, but they can be performed as often as daily (but can be monthly, every couple of months, and yearly). The required amount of fuel for the burns vs the size of the fuel supply (at least the fuel supply at the start of the mission) is low. North/South burns require the most fuel, so these are typically suspended first when fuel starts running low. This causes the satellite to start fluctuating its position North/South over a 24 hour period.
The Earth actually net loses mass every year via Hydrogen escape to space. In any case, the amount of mass the moon and Earth accumulate via dust and meteorites is an incredibly negligible amount versus the existing mass of either.
The reason the moon is (very slowly) moving away from the Earth is due to tidal acceleration. The tidal bulge that the moon's gravity creates on Earth is slightly ahead of the moon's orbit because of the speed of Earth's rotation. This causes Earth to exert a pull on the moon in the direction of its orbit, which causes a transfer of angular momentum to the moon, and increases the size of its orbit. Since we are losing angular momentum to the moon, days are getting longer as the Earth rotates slower.
Well, the Earth/Moon situation is due to a particular way to transfer angular momentum between bodies. There are other ways, and there are probably scenarios where a satellite capture could lead to a similar situation, but I can't really speak to that.
A bunch of the other satellites we can observe in the solar system, i.e. the gas giant systems, consists of many satellites in orbit around a planet. When there are many satellites, they can enter into orbital resonance with each other (and other perturbations), which can negate the effects of tidal acceleration and lead to other things such as massive amounts of tidal heating.
Also, the moon is in a prograde orbit, meaning it orbits in the same direction as Earth. That's what allows Earth to pull the moon in the direction of its orbit. If the moon was in a retrograde orbit, meaning orbiting against the rotation of Earth, it would actually be decaying the moon's orbit. Jupiter has a bunch of satellites that are in retrograde and have decaying orbits as a result.
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