1) It really depends on the crowds and the guards at the door. Sometimes they will let you in early but don't count on it. They are less likely to let you in at 9 as they will be to let you in early if your time slot is noon or after.
2) You won't be allowed to run inside the museums. You can go directly to the chapel (along with hundreds of others with the same idea) and then go back into the rest of the museums. If may be difficult with the crowds but it can be done.
3)The Vatican museums are crowded year round. It makes no difference. The galleries will be packed with bodies. The chapel will be wall to wall from open until close.
4) Unless you book an off hours tour you will run into hundreds of others who all want to see the chapel. You can rush directly there and backtrack but you won't find the chapel empty at any time of the day. Unfortunately, they don't offer the eveing visits to the museums past October or I would suggest that. Just take your time and realize that it's one of the most popular sites in Rome and that thousands of people go through there every day.
Donna