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The Space Shuttle was able to place a satellite into orbit and to return to Earth.
Is there any second stage that may deliver the payload satellite into LEO and do a deorbit burn after that using its RCS and remaining propellants? At least going to a very low orbit with a short lifetime of some months?
Not necessarily to reuse the second stage but to remove it from the orbit and to reduce space junk.
Yes. Several Falcon 1 and Falcon 9 upper stages, after payload separation, loitered for a few months in LEO.
Other upper stages have done more forceful deorbit burns, to disintegrate after only a few orbits rather than a few months.
To deorbit and land intact enough for reuse is much harder for a stage that's reached orbital velocity than for a booster that never got nearly so fast. Maybe the only "stages" that achieve that are crewed capsules, and even they (Apollo) jettison their engines before reentry, so by that point they're closer to payload than stage.
Yes, my belief is that currently nearly all upper stages return to Earth (burning up, not landing...). There isn't any law saying that they need to, but arguably there should be. But despite the lack of a law saying that they must, most launch providers want to play nice and not leave space junk in orbit if it can be avoided. This is usually done in a fairly controlled way so that the upper stage comes down over the Pacific where there will be little/no ill effect if parts of it don't completely burn up. There was quite a bit of scorn directed at China about a year ago when they had a big upper stage that made an uncontrolled re-entry. e.g. https://spaceflightnow.com/2021/05/10/nasa-chief-criticizes-china-for-uncontrolled-rocket-re-entry/
SpaceX seems to have been investigating having the upper stage of the Falcon-9 be recoverable. But they concluded that it wasn't feasible, or at least not economical because of the mass of the heat shield and the payload reduction that would result. Starship (if it works) will have a fully recoverable upper stage. It looks like New Glenn (if it ever flies...) might have a reusable upper stage.
