Stormtide Leviathan's second ability causes each land on the battlefield to have the land type Island. Each land thus has the ability "{T}: Add {U}." Nothing else changes about those lands, including their names, other subtypes, other abilities, and whether they're legendary or basic.

If Stormtide Leviathan loses its abilities, all lands on the battlefield (including those that enter the battlefield later on) will still be Islands in addition to their other types and will still be able to tap to produce {U}. The way continuous effects work, Stormtide Leviathan's type-changing ability is applied before the effect that removes that ability is applied.

Islandwalk cares about lands with the land type Island, not necessarily lands named Island.

Stormtide Leviathan's third ability affects all creatures with neither flying nor islandwalk, regardless of who controls them. They can't attack any player or planeswalker.