Academy of the Mad is a Support card that says: "Exhaust to lower the next Hastur character by 1". I'm playing is so it gets refreshed at the start of each turn, and can be used in every Operations phase to reduce the cost of a character, over and over so long as it is in play. Is that correct?

Oath Bound Servant says "Lower the cost to play Hastur by 1"

Also, is this a legal stacking: In the operations phase assume Oath Bound Servant is already in play, exhaust Academy of the Mad, then reduce the cost of Hastur by 2... one for Academy, one for Oath Bound.

Yes. That is correct.

That is how I read the rule.