Yes, it is possible to have pockets of air underwater as long as there is something there to contain the pocket.
You can easily demonstrate this by turning a cup upside-down and submerging it in water. If you put a napkin in the bottom of the cup before you do this, the napkin stays dry.
If you do the above experiment, but dive down in a swimming pool with the cup, you will notice that the air pocket decreases in size as you dive, and returns to normal size as you surface. This indicates that no air is being lost. Instead, the air is being compressed.
These observations indicate that the air pocket does fill part way up with water, but does not disappear. This is because as the air is submerged, the water pressure on it increases. This shrinks the air pocket's volume until the pressure in the air is the same as the pressure in the water surrounding the air.
Yes, you can breathe air in an air pocket - it's normal air. However, you will exhaust the oxygen supply quickly if the air pocket is small.