Why aren't nuclear reactors and spent fuel storage built below water level?

Gravity feed for emergency cooling of reactor IS a good idea and used in 3rd generation designs of power plant nuclear reactors; its called passive emergency cooling. However, this is done by means of tanks of water positioned above the reactor. And altho passive means of cooling is done automatically w/o aux power and w/o operator control - this is only possible for a limited time = 3-days before the tanks run dry and need refilling.

Reactors are built near large water sources (Oceans) but on directly connected to them. There is always a desalination plant built in proximity to filter and purify the water before it is brought in. And there is always a set of cooling towers and a retaining pool of some sort at the output to allow the water to cool to a normal temperatures after having been used to cool the plant.

Building under the water table near the ocean is both cost prohibitive and the plant is then under constant environmental risk from the corrosive effects of seawater.

You did happen to notice that the plant in Japan survived the earthquake with minimal damage. The problem is really the cooling of both the reactors and the spent-fuel containment facility.

I wish the media would do more to that the explosions have been hydrogen and not containment facility breeches.

You're missing quite a few things::

1- Building a reactor underground would be enormously expensive and add even more technical challenges to the already-difficult exercise of building and operating it in the first place. Maintenance, modification, and expansion would be extremely difficult.

2- The situation you suggest would flood the reactor entirely, which would be as much of a problem as the meltdown itself

That would risk contaminating the entire water table. You don't want to let radioactive materials loose in a major body of water, which is what you're connected to if you're below water level.