τί δὲ ποιῶν αὖ οὗτος ὁ παιδαγωγός σου ἄρχει;
this αὖ baffles me. Then, by doing what does he rule you?
I believe that it applies on the narrative level. He’s asking the question again, or at least a different version of it, because he wasn’t entirely satisfied with an earlier answer.
It’s a less forceful version of this, I think: “I ask again, what does he do…”
yes, this is probably correct, that is, but doing what (I am asking you) further/ again, does he rule?