Is there any air under water?
there are at the very bottom these big air pockets i think.
Yes, the "o" in h2o stands for oxygen. So, technically the water, itself, contains air.
Yes, although it is not in the form expected, it is in dissolved solutions. Water is made up of oxygen and hydrogen and without the presence of oxygen the sea creatures would die. For more details, check http://en.wikipedia.org/wiki/Underwater.