Wow, questions answered within a few hours.
@Yash. and
@Parth D for the answers so quickly.
1. Parth got it right. As long as the ball is in play, every run is allowed. There is an archaic rule that isn't covered in the official Laws that says you can't run more than a certain maximum amount of runs once a "lost ball" is called, but if you do call lost ball, the batting team scores the agreed amount of runs. Usually nowadays it's six, but it could have been any amount that both captains agreed to before the match. Up until that number of runs, the batsmen can run however much they please.
2. Yes, Tests against Zimbabwe (unfortunately) were official 4-day Test matches in the modern era. It never even made it to all four days however, as South Africa easily steamrolled the Zimbabwe side. Back in the day, Tests would be played over three days, or four days. Five days would include a "rest day" where players would telegraph back home and generally relax. But with Australia/India, Bangladesh/West Indies happening so recently, the talk of four day Tests has died for the time being.
3. Well...not really. Spider cam is a part of the official recording equipment agreed upon by both captains. A projectile thrown by a fan is not part of the game. If between you and square leg the ball was definitely headed for six, then six it is. (To any commentators reading this, I understand the need to call the sponsor's name. But it's SIX. Not A MAXIMUM. Okay? You can still call it a six. Please.)
4. I would talk to my match referee, who will talk to the director to ask to display a message. I will inform the fielding captain that I cannot make a fair decision until the crowd is silent, or at least quiet enough. I expect some noise, but not so much that my judgement is impaired.
5. This is why I love Aleem Dar. Talk done.