1) Yes, the team is allowed to build a bucket or other container for storing people/pets. The robot is allowed to push the bucket around the table, or the bucket may be attached to the robot. Both are okay.
However--a bucket that is not attached to the robot may incur a Junk Penalty (probably 5 points), at the end of the match. See Rule 13.
2) Yes, the team may manipulate items in Base by hand while the robot is out of Base. See Rule 21.
3) The team should be prepared to work around this (Rule 19). I cannot tell from the Field Setup illustration whether the bottle is standing up or not. The fuel can is definitely not standing up. The team may ask the referee to check and reload the truck, but may not ask for a "custom setup".
4) Tagging in and out may happen at any time (Rule 2). Different referees handle it differently. For myself, if one team member is leaving while a new one is coming in, then I think that's okay. I just don't allow three people at once up at the table doing stuff (or in position to do stuff).
5) The Challenge doesn't specify any dress code for coaches (although Gracious Professionalism discourages attire not appropriate at a public event with children.) Many coaches wear team clothing, hats, etc. to support their team, and some will be more costumed. As you note, it's not allowed for a costumed coach to be part of a presentation. I have it on good authority that one Tournament Director this week will wear a tutu, and may have on a wig or other headgear. Bottom line--yes, let your imagination run free!
VA/DC FLL Referee Advisor