- Assuming all laws of yichud are complied with, are there additional concerns arising from a male rabbi teaching Torah classes to a female student?
- Does the answer to 1 differ if they are both married, one married one not, or both single?
- How does one apply the mishneh in Pirkei Avos of not to engage in too much conversation with women to this situation?
Regarding question 3, I assume you are referring to Avot 1:5. The term used in the Mishnah is שיחה which loosely means "conversation" or "chatter". It does not refer to Torah talk. How can it refer to Divrei Torah if the end of the Mishnah says that one who engages in שיחה takes away from Divrei Torah? That would be paradoxical!
See Rav Bartenura's explanation to what type of "talk" this refers to. He mentions talking about bad things that occurred to him (a form of lashon hara) at the market or similar.
In short, the Mishnah in Avot doesn't seem relevant to your question.
DailyHalacha.com says that so long as there are two or more ladies in a room that does not lock, or with the door open if there is a lock, or if his wife or mother are in the room a man may teach ladies.
Thus there are no additional concerns besides the Yichud concerns.