Would the companions become Wardens?
Actually, given how much time they spend fighting darkspawn, I don't know how they managed to avoid Blight sickness at all. Consider this; every time a non-Warden character gets injured by a darkspawn, that person SHOULD contract the sickness instantly like Wesley did in DA2. So, especially for the melee fighters - Sten, Oghren, and Zevran, I'm looking at YOU - it should seem almost inevitable.
But, since it's somehow not, here's who I think would agree to join.
Morrigan - absolutely not. Not under any circumstances whatsoever. She'd die first.
Wynne - It would be madness to try it, even though she might be willing. Wynne is too old and she's not hardy enough to do this.
Shale - it wouldn't work. She doesn't really have blood to taint anymore.
Dog's a dog. However, this might be interesting, if it worked....
Leliana and Sten would have to be tricked into it. They've both got other things they'd rather dedicate their lives to rather than darkspawn fighting. These are two who might drink from the cup in order to be better fighters, and then just lose it when they find out that they are destined to become ghouls. I wouldn't want to be in the same room with the Sten when he experiences the nightmares and learns how short his life is going to be. That would not be pretty.
Zevran's already figured out that this is not for him.
Oghren, of course, was not only willing, but a nearly perfect choice.
But then we consider the question that raised this debate: why were they unwilling to recruit anybody else? How many soldiers and guardsman were in that room for the brawl that could happen if the Warden didn't win the initial argument? What members of Ferelden nobility might be willing to join the Wardens in order to save their terribly threatened home? What about Bann Wulff, who has lost his entire territory to the darkspawn? What about Teagen, who watched Redcliffe being decimated by a horror nearly as bad? And yes, what indeed about Cauthrien, who might well offer herself up in place of Loghain if it meant that he might be incarcerated rather than executed?
I think one of the reasons why the Landsmeet does not really work for me as an exercise in moral choices is that the most sensible possible choices were never even considered. The choice we were given was that EITHER Loghain must be executed on the spot, and Alistair must then become King, OR Alistair must be thrown in the garbage can. It's quite obvious to me that Loghain should have been sent to prison while Riordan allowed the Warden to offer the chalice to somebody else who didn't have quite such a history of brutal hostility toward the order, and the question of who should reign should have been a separate issue altogether.
I will say, quite honestly, that I think executing or exiling Alistair would be not only stunningly unjust for what is really little more than a single angry verbal outburst, but, excruciatingly stupid. When you have only three Grey Wardens in Ferelden you don't let one walk out the door and you don't let Anora chop his head off so that her political power will be more stable. You don't allow a preventable loss, period, because at that moment, killing that damned dragon is much more important than anything - ANYTHING - else going on. It's much, much worse than what happens if Bhelen takes the throne, because Harrowmont doesn't have a particular, very rare medical condition that is absolutely vital to the survival of the entire kingdom. I have heard people argue that Anora is right to execute Alistair because rebellions can be raised in his name, but it just doesn't wash at all when he's one of only three people in the entire country who can perform the only possible task that can keep Ferelden from being as ruined as the Anderfels.
The only excuse I can give for Anora even suggesting the execution is that she does not know that the man who kills the Archdemon dies. (Heck, if she knew, she might be very helpful about pushing Alistair right into the glory seat! He's a hero, she's the queen, Ferelden is saved, everybody wins!) If there were any chance at all that she knew the Warden who killed the archdemon would die, (because after all, she knew that the Joining could be fatal, and that was supposed to be a secret, too) then the very fact that she wanted to execute somebody who could save the nation would, by itself, render her completely unfit for the crown as far as I am concerned. Nobody should ever put their own political power ahead of ending the Blight.
Of course, nobody should stomp off in a huff because his worst enemy didn't get splattered across the floor, either, but we know for sure that Alistair didn't know the numbers game. I will assume that Anora did not, either.