1) Most skill rolls are made when you start the action. The reason for that is that the time to perform a skill roll can be different depending on if you succeed of fail with the skill roll.
2) How long it take is one of those GM discretion tings. I my campaign I mark off (I have a calendar) x number of days if the players are working on a project. How many depends on what it is but usually anything from a couple of days to a week.
I should say that we have our assembly meetings at the end of a session and I draw a threat card also at the end of a session. This way I have more time to come up with events that can happen when the players are working on projects and I have more time to prepare a better story for the threat card.
3) If there is no assembly or no one in the ark to work on a project, the GM can start a project and also can deduct WP equal to the number of players for a project of his choice (p 103). So this means that there is progress for the ark even though there are no players there. It is in the hands of the GM. A perfect opportunity for the GM to stir things up in the ark.
In one campaign, my players where gone for several sessions (7 I think it was) from the ark (there where on an expedition to investigate a possible bunker). There was a conflict in the ark when the players left so during those sessions where the players where gone and I as a GM controlled the assembly. I played on this conflict. So when the players finally got back, they found that the ark had finished a militia, tribunal, torture chamber, surveillance and the project autocrazy was started but not finished. The reason that I picked those project was that the ark had been attacked with the help from people from the ark so the people wanted protection. The players found themselves with a new problem. A boss that hated them was in her way to become a dictator.
4) The body count is done in the end of a session so whatever has happen in the session counts towards this.
5) When I ran this event I decided where the cure was to be found and I calculated how long it would take to get there and back. It would take two days so I decided to have an additional d6 body count deduction at the end of each session for each game day that has passed. It took the players three days to get the cure so 3d6 mutants was lost due to the fever. I also decided when this threat started that if the players didn't find any cure after one week, a group of NPCs will find a cure instead (i though that loosing an addition 7d6 mutants was harsh enough).
“I’d say thistles, but nobody listens to me, anyway.” - Eeyore