I never studied these specific classes, but my immediate intuition is that an n-input fan-in AND or OR gate can be reduced to a tree of 2-input gates with depth O(log(n)), which preserves polylog complexity, so surely AC = NC.

Wikipedia agrees :)

If you specify the exponent of the log, you get a different answer.

Yes.

There is a beautiful proof of the disjunction between AC0 and NC showing parity cannot be done in AC0 using harmonic analysis of Boolean functions