With War! in print, I've decided I'm going to tackle that, and a few other things that have come up, and release an updated FAQ sometime in the near future.
A) No, you cannot spend downtime before your first mission. In theory, Week One on your calender should be your first Missions adventure.
B) Yes,. you can, again in theory, spend an unlimited amount of time in Downtime between Missions, so long as your rent and other upkeep is paid. This may be addressed though, because as folks like to point out, this can be abused heavily.
C) I missed something in the FAQ. A couple somethings. "Item Creation" of all sorts should be off limits to PCs, mainly to prevent abuse. Creating FOci was covered, but I left out creating your own spells (Which, since it requires GM approval, is out by defauylt anyway, but never hurts to be clear) and Programming. Mainly, this is because the price associated is one major balancing factor. And again, as has been proven by the number crunchers here, you can basically take nearly unlimited amount of free time for almost no money, which basically means you simply go "I have free shit and you don't, neener neener". It's not really fair to the sammies, mages, and riggers who can't create their own cyber, foci, spells, and drones for free.
That and I've always been a fan of pointing out that if you're going to make your own items, why the hell are you shadowrunning? At that point, open a shop and sell your shit and make the same money or more without the risk of getting shot.

I should add that in a home game, the GM gets to set certain balances on the game by controlling downtime. Things happen on his timetable, and the world turning around you. Enemies won;t wait for you to code things. Important jobs that you can't afford to turn down come up. Things will interfere, when they interfere, if the GM doesn't want to let you spend infinite downtime coding and whatnot. Missions doesn't have that luxury, which is why I note that cost is one of the balance factors.
Bull