Question - once I've completed a story, do I have to complete the rest of the mission to get awarded for it, or can I just exit to the menu?
You do have to complete the mission to get all the challenges officially 'ticked off' the list, but they also register immediately in terms of receiving XP and (I believe) unlocks - even if you die or reload.
This means that when you get to a point where you're cleaning up a few stray challenges, you can, let's say, poison Silvio, then immediately reload a save and shoot him in the head, then reload and push him off a cliff, etc. and once you complete the mission, they all get there ticks in the challenge menu.
Hey I'm sure this thread is the same questions over and over, but I didn't finish Hitman 1. I just bought Hitman 2: Gold Edition. Can I pick it up there or should I actually re-download and finish 1 before importing it into 2?
Just do it in 2. You lose the progress you made in the first game, but you get all the upgrades from H2 in H1, like being able to hide in foliage, the briefcase, etc. Plus, progress made through the Hitman 2 client (including the old levels) WILL transfer over to H3.