Are there any underwater natural disasters?

Mud flows (landslides underwater) happen a lot. There are also anoxic events (oxygen contents go to almost zero), and there are erupting volcanoes underwater in many places. We don't call them disasters for ourselves because we don't live there, unless the disaster causes a die-off of a fish resource. I suppose we might call it a disaster when thousands to millions of dead sea animals wash up on shore, too. But these events are pretty disastrous for whatever is trying to live in those places whether or not they affect humans.

Yes, many. Earthquakes and volcanoes for instance.

Yes. Large Undersea Earthquakes that cause Tsunamis.
Don't you read the news? Large Tsunami hit Japan and killed thousands of people and damaged a Nuclear Reactor. (a few years ago)

Dead zones are hypoxic (low-oxygen) areas in the world's oceans and large lakes, caused by "excessive nutrient pollution from human activities coupled with other factors that deplete the oxygen required to support most marine life in bottom and near-bottom water. The area of temporary hypoxic bottom water that occurs most summers off the coast of Louisiana in the Gulf of Mexico is the largest recurring hypoxic zone in the United States. The Mississippi River, which is the drainage area for 41% of the continental United States, dumps high-nutrient runoff such as nitrogen and phosphorus into the Gulf of Mexico.

Tsunami that killed 100,000 people in India Christmas Eve

Right now Venezuela is missing a submarine, so yes