Do underwater volcanoes affect sea level?

I had a thought while watching this video of an island being created...
www.youtube.com...

There are a bunch of different ways to ask this, and even more variables, so I'll try to keep it as simple as possible - short and long term scenarios

Short term - The material they eject displaces water, so it seems possible that if (hypothetically) there was a gigantic eruption, and a new island the size of texas was created over a few days...

There'd obviously be tsunamis and all that, but when the water recedes, would it have somewhere to go?
What replaces the material underneath the volcano?


Long term - Kind of the same question I guess... Does the water have a place to fill in under/throughout the volcano system, or will sea levels continue to rise tiny tiny amounts daily as undersea volcanoes continue to exist and do what they do?