Can I use the movement ability of Ship/Lighthouse to move through a water hex adjacent to a warrior? (It is clear that the movement mustn't end there, I am only talking about moving through such a hex.)
You can't move through/to hexes adjacent to a warrior.
The rules are not as precise as would be ideal, but the rules for one ability can never take into account all other abilities. The warrior makes the adjacent spaces into obstacles; that prevents many abilities from using those spaces.