I am wondering what is the tallest fully underwater seamount in the world ? When I say the tallest, I mean the one that is the nearest of the sea level. EDIT : The highest that I found would be at about 50m below the water's surface do you know any other seamount that would be even higher ?
According to the Oxford Dictionary of Earth Sciences,
seamount Isolated, submarine mountain rising more than 1000m above the ocean floor. The sharp, crested summits of seamounts are usually 1000-2000m below the ocean surface. Seamounts are of volcanic origin.
To avoid classifying seamounts by arbitrary sea level (dependent on availability of surface water), the key point is then that seamounts are features of volcanic origin that rise over 1000m above oceanic crust.
The Hawai'i-Emperor seamount chain is of volcanic origin; all of the islands in this chain are seamounts. Mauna Kea only rises 4207m above sea level - but measured from its base on the oceanic plate it is 10100m high, much taller than Mt Everest. Mauna Kea is - pretty conclusively - the highest seamount in the world. The Woods Hole Oceanographic Institute agrees.
This answers one of the questions you list, but the shallowest seamount, per this list on Wikipedia is Banua Wahu, which is 8 meters below the surface, but has risen above the surface and fallen below the surface several times in recorded history.
However, this is definitely not the tallest seamount, standing only 400m high from its base.
A larger database by Yesson, C et al. includes several seamounts that are within 2 meters of the surface. However, the data does not suggest when the measurements were taken (high tide or low tide), which would have obvious implications. Also, there are many seamounts within that range (I stopped counting at 100).