Can you breathe with a drop down mask under water?
No, you cannot - at least not with the default masks.

In principle, the oxygen systems on board can all provide at least 15 mins. of oxygen, in some of the worst conditions (indeed, that's what they're designed to do). However, their sole purpose is to make high altitude air breathable. The key principle here is partial pressure. Basically, the partial pressure of a gas is the fraction of a certain gas in a gas mixture times the ambient pressure. Since normal air consists of about 21% oxygen, the partial pressure of oxygen at sea level is 0.21 atm (atmosphere). If the partial pressure of oxygen is too low, your lungs actually lose oxygen to the atmosphere when you breathe - much like a deflating balloon.

When the cabin pressure drops, the partial pressure of oxygen drops proportionally. In that case, oxygen masks are provided. They do not raise the pressure of the air you breathe in, but raise the partial pressure of oxygen by filling your mask with oxygen. If the cabin pressure would drop to 0.2 atm, and you would be given 100% oxygen, the partial pressure of oxygen would be 0.2atm, which is about the same as at sea level - perfectly survivable. The mask as such does not need to be airtight - it just needs to make sure that there's enough oxygen flowing into your mask to displace all the useless nitrogen in the air. Indeed, the mask you will see on airplanes are generally flimsy, plastic masks.

Under water, these masks are not sufficient at all. They will probably turn into a frothy, bubbly mess of a water/oxygen mixture, especially when the aircraft is filling up with water, due to strong currents. So, you would have to wait for the currents to diminish, and then hold your oxygen mask in such a way that no water will enter it - this probably means you will have to face upwards all the time.

Of course, you can get creative (pull the mask of and breathe from the tube), or luck out with a high quality mask that happens to work underwater. But I'm afraid that as soon as your plane is sinking, it's too late to play the hero, and your best bet would be to get out and at least save your own life.