Starks have Sansa and Cat out. Sansa has a 'kneel card to save attatchment'.

Starks lose a military challenge. Can the defender choose Sansa to be killed and say "I choose Sansa annnnnnd kneel this card." Or does he have to choose a card that can actually die first?

When something leaves play it goes through an area / process called moribund

it has a pre-defined destination - e.g. Dead pile, discard pile, return to hand - but that does not happen until all passives and responses have completed

Essentially if someone is killed and is standing at the time they are killed, they can still be used, i.e. kneel in moribund to pay a cost, trigger an effect etc. ... in other words in behaves much the same as if it was still in play at that stage

Dale: The claim for a military challenge is considered a kill effect. If you have a "save from being killed" effect available, you can use it to save the character you choose to die for military claim. So yes, there is nothing saying that a character MUST die for military claim and the defender can avoid the consequences of military claim with save effects. (It's also worth noting that Sansa can actually die in your scenario; after all, you could choose not to trigger the save, or your opponent could cancel it - and then she would die.)

Hast: The moribund rules really don't apply in this scenario. The whole point of a save effect is that it stops the card from ever becoming moribund in the first place. If Dale chooses Sansa for claim and uses Nymeria (I'm assuming) to save her, nothing goes moribund here at all.