Well, this ain't a 100% save, I'm missing two gallery entries. One of them requires finding two more quiz coins (you need 38) which I honestly can't be arsed to do, and then approaching the third 'quiz daughter' (it probably was translated from 'quiz musume' which really should mean 'quiz girl/maiden', but I digress). The third 'quiz daughter' can be found by going to Ilets city, taking the free-roam ship, and going to an island to the South-West. The other missing entry is a mystery to me.
For anyone who can be bothered to continue where I wouldn't, no-encounter has been equipped and the quiz coins appear as a sparkle on the ground. Good luck!
Some loose ends: I don't know what the Mysterious Lamp is for. If you manage to figure it out, good on you. There is also an infinite maze thing at the final island found at the far North-West of the ship free-roaming area. I haven't been able to crack that one either. Also, the secret password to input in the Secret Room? Mystery to me, but it might be because it's expecting Jap characters input.