According to my understanding the aforementioned verbs have two related meanings like the English word 'to leave'.
a) to misplace. This makes sense somewhat because in order to misplace something you have to have put it someplace else. The prefix 'о-' meaning 'around/about' may have a sense of approximate (e.g. 'I approximately placed my keys somewhere over...')
b) to depart. This meaning does not make sense. If you are departing from someplace, why would you use the verb оставлять? You know where you are going, there is no sense of approximation. What are you putting away? Is using this verb wrong when referring to people, because it would seem more fitting when referring to objects only.
c) Why would оставлять and оставаться be antonyms? They have the same prefix and the same root. Why would they have contrasting meaning and what is their morphological relationship with the verb ставить?