Have you noticed that you can put a metal needle to float upon the surface of water, although if you push it a bit down, it will inexorably sink to the bottom?
In the interface of two fluids like air and water, there will be an ultra-thin layer of molecules that align themselves into a membrane.
This membrane will be able to exert a force we call "surface tension", which is the "culprit" in why the needle floats. If you add a bit of soap in the water when the needle is floating, it will begin to sink, because the soap destroys the watery membrane the needle rests upon.
As for why the air-filled bubble isn't squeezed to death, consider that as it becomes compressed, the interior pressure will INCREASE well beyond the ambient fluid pressure.
Thus, you will reach an equilibrium, in which the surface tension and ambient fluid pressure will try to collapse the bubble, whereas the interior pressure works in the opposite direction.
The sphericality of the bubble is mainly due to the tiny size of the bubble; the magnitude of the pressure doesn't vary much, either the internal pressure or the external pressure.
Thus, the pressure is equal in all directions, favouring the (almost) spherical shape.