What is the ruling on naming the one who recites the daily Quranic litany to you in the home as "the Quranic female teacher" or "the Quranic female companion"? If it is forbidden, what is the reason, and does this differ from "the Quranic school"?

The Shari'ah rule does not prohibit titles and names, as long as they do not include a Shari'ah violation or corruption. Attributing the title of "Mu'allimah" (female teacher) to the Qur'an has an apparent appropriateness due to her occupation and specialization in it, similar to attributing the title of "Nahawi" (grammarian) to grammar or "Usuli" (jurist specializing in Usul al-Fiqh) to Usul. The same applies to describing a memorization center as a "Qur'anic school," so there is no harm in this attribution.