Specifically, my opponent won a military challenge with Defenders Khal Drogo in which he was blanked until the end of the challenge by Frozen Moat. Does his opportunity to use his "after you win a challenge" ability kick in before or after Frozne Moat wears off (i.e. the end of the challenge)?

Nope, the challenge ends after all the responses while putting all the moribunds cards in the dead/discard piles.

Apllying the claim is only the resolution of the challenge, not the end of it at all.

So if the Khal is blanked with frozen moat, he won't be able to trigger his ability.

Effects like Deadly and "A Lannister Pays His Debts" are a good way to remember this. Both activate after you resolve the challenge (determine winner, claim, unopposed, renown, that sort of thing), but both also refer to "participating" characters. If the challenge were technically over when passives and Responses to the challenge were being resolved, Deadly and ALPHD would never have legal targets.