How does the rotating wheel overcome gravity, since while whirling, its axis is leaning and it is underpinned only in one point?

The whirling wheel is in this case a gyroscope, and the explanation of the phenomenon is precession: If turning-moment coming from gravity is perpendicular to the turning (and the turning vector's) axis, then the resultant turning-moment will be perpendicular to both vectors. Thanks to the permanent moment coming from gravity and the drag, the speed of rotation is slowing down and the speed of precesson is growing. Growing lasts till the moment when affecting forces are not enough for supporting the weight of the wheel, this is when precession ceases and the wheel falls down.

The most famous gyroscope is nothing else but the bycicle, whose stability is produced by its own wheels! It is more stable if it moves with greater speed. If it stops, it falls to the side!

The frisbee, the spinning-top and the yo-yo function according to the same principle.