1. Yes, the team may remove objects from Base/Safety while the autonomous Robot is out in the Field doing other stuff. That's because the team is allowed to handle objects that are completely within Safety.
2. This is both an easy and a complex question. No, R12 does not allow the Robot to Launch if any of it is sticking outside Base. There is no official Penalty, other than that the robot can't start, or if it does start, then by R10, it's not allowed to touch anything.
So, when looking at FLL Core Values and rule GP1, that's kind of harsh treatment for a team that didn't fully understand the Rules, and can't really make any changes on the fly at a tournament. We don't inspire the participants and advance the goals of FIRST by making a team stand around unhappily looking at a stationary robot. There are a couple of ways that your tournament may handle things. I prefer letting the team run their robot, and having a chance to show what they have been working on. The bottom line will be that the team won't be able to receive the Robot Performance Award based on the results of improper launches during matches. (And the FLL Core Value "What we discover..." and GP1 bullet 4 say that we all acknowledge that that is okay.)
3. As has been said, most tournaments distribute draft schedules about a week in advance. There will be fixed judging times. My recommendation to all teams is that, if things are running late at a tournament, then always go to your judging session if it conflicts with a table round. It's way easier to reschedule a table round than a judging session. For what it's worth, I know that the Fairfax tournaments haven't sent schedules out yet, because they are still being worked on.
Virginia-DC FLL Referee Advisor