What would happen if you submerged a stargate in the ocean and either dialed out or someone dialed in?

Immediately after dialing? the water occluded by the Kawoosh would be vaporized. After that, not much. Its pretty explicitly stated in an episode (don't remember which) that the gate's event horizon is used to filter out what does or does not pass through the gate. It effectively limits it to only solid matter with momentum, or energy, to pass through it. That's why you can have differing atmospheres, or even a gate in hard vacuum without any negative effect to the opposite side.