I think they'd adapt OK, especially since you mentioned using a sunroom or basement for the rest of that first winter.
I think moving them cross-country would be a nuisance, but doable if it's important to you.
If it's important to you that these particular chickens be loved and cared for during the rest of their lives, I would take them with you.
If you don't care about these specific chickens, then I would eat these ones, and buy new ones at the new home. (It does not look like this would fit your situation, but I mention it in case it does.)
For the bonus question, about the rooster: move first, then decide! If you do get more chickens to add to these, then of course you'll be dealing with integrating various flocks at some point, but that will work better if you're moved and settled before you try it.
I think moving them cross-country would be a nuisance, but doable if it's important to you.
If it's important to you that these particular chickens be loved and cared for during the rest of their lives, I would take them with you.
If you don't care about these specific chickens, then I would eat these ones, and buy new ones at the new home. (It does not look like this would fit your situation, but I mention it in case it does.)
For the bonus question, about the rooster: move first, then decide! If you do get more chickens to add to these, then of course you'll be dealing with integrating various flocks at some point, but that will work better if you're moved and settled before you try it.