@K311Y,
All but the Faerie items are the Neo id numbers. To check: type the item into the shop wizard, scroll over (do not click) a ship listing, look for "obj_info_id=###" in the bottom of your browser. The number following is the id number.
The Kadoatie and Sword numbers are from Jellyneo... which is sad because those numbers suit the pattern. I have a feeling that maybe you don't need to know the id numbers of them, but rather they serve the purpose of being a clue to a Faerie answer.
ids: watermelon 36887; bed 33327; PB 23673; plushie 36956; novel 34832
bucket of sand (relevant?) 36762