Do airplanes stay still on the air?

Generally, no. But if there is a headwind sufficiently great, an aircraft flying into it will have a zero or actually a negative ground speed. You can see how this works if you consider a light plane, as light aircraft are generally the ones most affected. If a small plane can only fly at 80 miles per hour and there is a strong, 80 mph wind aloft, that aircraft won't "move ahead" flying into it, and for obvious reasons. It is a documented fact that the military kept track of jet streams during World War II. The had to. Those jet streams, the high altitude, high speed moving "rivers" of air, could stop the old propeller driven bombers in their tracks. Yes, they'd be flying, but could "hover" over a spot on the ground when flying into headwinds like that. Their ground speed was nil under those conditions.