Is someone who took a shoe, thinking it was theirs, then doubted that, required to do anything, and are they considered a thief?

If you find a sandal in a place other than where you put your sandal, and you suspect that it is not yours, then you are not permitted to take it, because it is considered a lost item (luqatah) and must be announced. However, if you do take it, you must announce it immediately, because its owner might despair of finding it if you delay in announcing it. And if you delay in announcing it, you do not own it after the passage of the one-year announcement period; rather, it remains a trust with you, or you hand it over to the public treasury (Bayt al-Mal).