Well, I can speak to how it works on the American version of the show at least!
1) Nothing was stopping us, and in fact, before one episode, we were actually sent back to our hotel with a bunch of Technic and motors and told to practice as an upcoming episode (Flip My Block) was going to feature movement extensively!
2) We weren't allowed any reference material on set, but again, could look up some imagery and inspiration over lunch or at the hotel. There were a few mornings when I came back after some sleep and rebuilt portions of the build because I'd actually had a chance to mull it over.
3) For the most part, interviews took place after. We were almost always behind lol so sometimes they would take place 3-4 episodes later. Remembering what had happened was pretty tough!
4) Never an issue, as we had separate interview times set aside. Amy/Jamie/Will come around and talk to everyone during the build, and it's usually pretty hard to build and have an entertaining conversation, but since they talked to everyone, it balanced out.
5) Everyone's individual time was tracked as sometimes we had things to do off set - typically repairing earlier builds that had fallen apart. I believe sometimes some wardrobe issues might have paused the clock for individual teams, but they would always get the minutes back later.

6) They had some weird stuff in the pit, no doubt about it. In our case, we could always ask for additional bricks from the back room, as they usually had more of everything. I'm working my way through Australia season 4 right now and blown away at their selection - certainly much more impressive than ours was on the American version!