Seems to come down to a fundamental misunderstanding of the game and using words incorrectly here.
Your character is a person - the one you select when logging in - and they are always the same character no matter how much you change class/job by equipping a different weapon.

No damage has been done to your character and they are not unable to gain experience points or increase level, you're just misinterpreting the information the game is telling you.
Equipping or removing a job stone will switch between a job and the class it is based upon; think of the job as the advanced form of that class. Whether you use it or not you still receive class* exp as they are one and the same, they grow together as one. Black Mage = Thaumaturge. Summoner = Arcanist.

Your character isn't damaged if you misplace an item, causing the error message that item cannot be found. Simply update the gearset and save it again so that it's no longer existing find an item that isn't there. You can avoid accidentally selling or depositing items which are registered to a gear set because they will have a mark in the corner of their icons when viewed in the armory chest to show they are being used in your set.