I have two cards:

Bastard Daughter in play which states: "Response: After you lose a challenge, discard Bastard Daughter from play to put a unique Sand Snake character into play from your hand or discard pile."

and

Dorea Sand in hand which states: "Response: After you lose a challenge as the defender, until the end of the phase Sand Snake characters you control gain: "This character does not kneel to attack."

I loose a challenge and discard Bastard Daughter to put Dorea in play. Can I use her Response as well, as it looks for me that action window is still opened.

It is still open, so you can do that.

But if it was a MIL claim, you cannot have chosen the Bastard Daughter for claim & then also discard her to put Dorea Sand in to play.

Slothgodfather said:

Indeed, per the rules on Moribund. But you could choose another character to die (or save the Daughter), then discard the Daughter.