When exactly is a player determined to have "won" or "lost" a challenge?

If a character says "Afer you have lost a (X) challenge, do (Y)"

Am I considered to have lost the challenge before or after I satisfy the claim for that challenge? If I want to (or am forced to) do effect (Y) of the card text, will that trigger even if I wish to kill that character to satisfy a Military claim?

There are a bunch of other threads on this. It all comes down to when passive effects and Responses are played during the resolution of a challenge.

So we fall back on the same things we always fall back on with framework action windows. You have to completely resolve all framework events (Steps 1-3) before you can initiate any passives or Responses (Steps 4 & 5) to any of them. So, even though you "win" the challenge before settling claim (you have to, or else you don't know if claim is resolved), you still have to settle claim, award unopposed and award Renown before you can look at passives or Responses to winning/losing the challenge.

So if the last card in your hand is A Lannister Pays His Debts, you are going to lose it for claim before you can play it as a Response. And if you lose a Deadly military challenge, you can choose the only defending character for claim before you resolve Deadly.

It's the same reason you don't get the card from Golden Tooth Mines ("after you reveal a plot card") until after First Player is determined and all plots are resolved.

Do take a look at the flowcharts included with the FAQ, if you haven't already. They are really useful for timing issues.