I have red viper in play and I lose a military challenge. I choose to kill red viper for claim and play No Use for Grief to search for Dorea Sand and put her directly into play. When she comes into play, can I immediately trigger Dorea Sand's ability or do I have to wait until I lose another challenge? (her ability says "after you lose a challenge as the defender ...)

Thx for any clarification you can give me.

You can trigger Dorea, and if you have Nest of Vipers in your hand at the time you can use that as well. Unless something goes terribly wrong, your army of Sand Snakes will now handily win you the game.

When you have multiple responses to make, you can pick the order to do them. If after completing one response you now have the opportunity for a new, valid response you didn't have before -- because you've just vomited out every Sand Snake in the game and Dorea is now in play, or your response includes draw and you get a card you can now play -- you are free to do so.