Cat girls appears after you lose some battles and will reset your stats if you agree with her. She will disapper after some time, IIRC its around when you got the quest for class change.I can't remember but the cat girl appeared pretty early for me, though that was the only time she appeared. Once I agreed to have my stats reset, she never appeared again.
As for crafting golden apples, after you land on the floating castle.....the guy near the ship will give you the option to craft them. You need 5 of the crystals needed to upgrade accessories and 1 black jewel dropped by girl number 39 and 40. He can also craft the skill reset, but honestly paying the guy for the stick is cheaper.
Im no sure on the first one but I think its the turtle one. You need to talk with a rat on the sewers, he is on the first floor, as soon as you enter go up across the bridge. Then you need to kill an enemy at the 3rd floor in the undersea cave, she is on the left side of the map,i need help cav anyone help me with three quest ? i stuck .and can't find how to do thiem.
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Second one you need to talk with a NPC in the first town castle, he is at the bar on the right. After that you will need to talk with 5 npcs that have a TARGET above them, there are 5 of them 1 on each city.
Last one you need 5 itens that drop from a mob at the desert dungeon, she is the one with a katar wepon.