What is the ruling on the fast of one who swallows saliva that flowed from his mouth during sleep while he is aware of it, and does the fast of one into whose belly something entered unintentionally due to obsessive thoughts become invalidated?

Intentionally swallowing saliva after it has exited the mouth invalidates the fast. However, it is unlikely that a sleeping person would do this intentionally. If it happens from a sleeping person unintentionally, it does not invalidate their fast. As for the saliva that entered your sister's mouth unintentionally, it does not invalidate her fast, because she thought it was permissible to swallow.