How did this shaykh, who is a mosque imam, come to know about the matter of sorcery without personally seeing the questioner, but rather relying on the names of the questioner and her parents? And is this considered charlatanry? And what should the questioner do about that?

There is no indication that a person can know remotely that another is bewitched, unless that person is a soothsayer or a user of jinn. Among the signs of a user of jinn is their asking for one's name and father’s name. A Muslim believes in magic and its effect, by Allah’s permission, but does not believe everyone who claims to have such knowledge. A Muslim should fortify themselves with the prescribed remembrances (adhkar), authentic incantations, and legitimate ruqya, and seek help through prayer and supplication.