when you get item to swim you can go directly to dungeon 3 following this way.
after you enter dungeon 3 it unlock a lots of trigger. Some here:
All boss before dungeon 3 are dead and reward chest waiting for you
Dungeon 2 door is open and don't need any music
Bridge to dungeon 2 is open and you don't need bow
as the main quest is update you don't care if you have only 1 key part, lily can open the minotaur door