What is the ruling on dealing with a Christian teacher who greets with "peace be upon you" (as-salamu alaykum), swears by the Qur'an and the Prophet, and wishes to see religious chanters (munshidin)?

We do not know what the questioner means by "doubt" concerning this teacher. However, in dealing with her, it should be observed that she is a woman, necessitating lowering the gaze, and that she is a Christian, thus she should be invited to Islam with kindness.