Unless a camera records gravimetric or gyroscopic data to the file, this would be impossible, so no doubt why it has never been done.
That's not needed for stabilisation. Maybe it could make digital stabilisation during recording better, but for stabilisation in post you will "just" have to look and compare the previous and upcoming frames of the video to stabilize it. Of course thats also complicated but as it can be done in video editing tools, there is no reason why a video player shouldn't be able to do it.
But I don't know any player that does.
That's what Mercalli and others try to do, but it just doesn't work very well
It depends how much you use it in my opinion. If you overdo it then yes, you can spot it on the edges of the frame, or you will have to crop a lot from the video.
If you have gyro data you will still have to crop and move the video frame and/or "guess" how the image area outside the frame looked. Maybe you will need less processing power if you have gyro data, and you stabilsation will work more consistently in scenes with lot's of movements on the frame - besides the camera movement itself.