At what point does Ellaria claim a power? Is it after renown?

She doesn't technically claim power but yes, it's after Renown is awarded. Claim, then Renown, then Passives, then Responses.

Yes. Well after.

She is a standard Response to losing a challenge. That means you use her ability in Step 5 of the "Resolve Challenge" framework action window. Renown would be claimed in Step 3. (If you want to get really precise, Renown is claimed in Step 3d of breakdown to resolving challenges provided in this thread .)

One of the greatest things she does is steal 1 power just claimed for Renown by an opponent. The combo of that with Vengeful is pretty darn potent.