On the second floor of the castle town inn, you can walk through the left wall in the room with the NPC that asks if there should be another room on that floor. The saw is in that room.
And yes, to get to the Lost Halls you need to play the whistle in a room with glassy tiles. Such rooms exist on floors 3, 5, 7, and 9, though you need to finish a trading quest to access floor 9's lost hall.



