I won't spoiler you too much, but suffice it to say that Tribunal's main quest is not open-ended at all.....you're given very little choice (well no choice really) in the path you follow. Just keep doing Almalexia's dirty work for her (I know, it's distasteful but it's the only way to finish the game) You're in for a surprise at the ending, I think
I have no idea what would happen if you went back and turned off the ashstorm machine....you could try it but I don't think it'll have any effect on the eventual outcome, which, like I said, is fixed no matter what you do. Almalexia won't send you to turn it off, as it happens...someone else (not you) has to stop the storms in the end.
Also, there's no way to avoid killing the missing Hand, unfortunately. As for Helseth's quests, that part confused me too....I kept running back to him with more info, only to be told to come back when I'd found out more. You just have to follow Alma's quests and when you've done them all and resolved everything with Her, go back to Helseth and talk to him. He won't actually have anything more for you to do, but talking with him sort of wraps everything up.
BTW, what did you think of Helseth? Foxy isn't he?