If i attack my opponent with Daenerys and name a character to kill after declaring attackers and my other opponent decides to support what happens to the target i had selected. Does the response still take effect as the response set up a lasting effect ?

Thanks

Also assuming it is a lasting effect, even if Daenary is killed before the challenge resolves, if the attacker still wins the challenge the target is stilled killed?

When redirecting with the Crown Regent, I assume that occurs before Daenerys's response ... though I probably would not be able to insist the challenge cannot be redirected if the target has already been chosen

Once the target has been chosen, the only way to stop the kill effect is by winning the challenge on defense. The play restriction is simply "choose a character without controlled by the defending player". Once that target is chosen, it does not matter if the controller of the character stops being the defending player or if that character receives any attachments before the challenge resolves. The kill effect will happen if the attacker wins the challenge.

By the time the other opponent jumps in, supports the original defender, and becomes the new defending player, the character is already chosen as that's the only time those play restrictions matter. Once you win the challenge as the attacker, that character will be killed(unless saved).

Also assuming it is a lasting effect, even if Daenary is killed before the challenge resolves, if the attacker still wins the challenge the target is stilled killed?

When redirecting with the Crown Regent, I assume that occurs before Daenerys's response ... though I probably would not be able to insist the challenge cannot be redirected if the target has already been chosen

Correct on both counts. Once her Response resolves, it doesn't matter what happens to Dany or her text (blanking, for example). The kill happens if the attacker wins the challenge.

And yes, if the challenge is redirected with Crown Regent, the identity of the defending player changes before her Response is triggered, so she has to choose from among the characters controlled by the new defending player.

Note that if the target has already been chosen, you would be able to insist that the challenge cannot be redirected (by Crown Regent) because the person with the Title missed their chance (assuming it wasn't skipped - eg, "I attack with Dany and choose your Robb Stark for her effect." "Uh, Dude, you jumped the gun. I have to decide whether or not to use Crown Regent before you can trigger her Response."). It's too later for that redirect -- although Varys, Lord Commander and "Supporting" could still make the redirect happen.

Thanks ktom ... the redirect is one area where our game can be very sloppy wit regards to timing

e.g. someone is attacked, the targeted defender see all these player actions being done, before defenders are declared, and then chickens out and wants to redirect