Disabled access to St. Peters and Sistine Chapel

There's a lot of information on the Internet, so I'd suggest you do some searches. It also looks like there are some companies that specialize in handicap tours of the Vatican.

Here's a starting place, with the information about the museum from the Vatican website and information about the Basilica from another website.

Museum info:
http://mv.vatican.va/3_EN/pages/z-Info/MV_Info_Servizi.html

Some Basilica info:
http://www.freepressindex.com/article-vatican-saint-peters-basilica-wheelchair-access-tips-and-reviews-by-sage-traveling-424089.html

FYI, the Sistine Chapel is within the museum complex.

One question about your plan to attend Sunday mass in St. Peter's Square. Are you going to be there on one of the few occasions when there is a Sunday mass in the Square? If so, you might want to check to see if tickets are necessary. Most Sundays, what happens is the Angelus, not mass. There are many masses within the Basilica every day.

EDIT: One additional point – the crowds being drawn by Pope Francis are huge. Consequently, I would suggest you preplan as much as possible, and don't necessarily expect cabs to be able to get you close to things. I'd suggest getting as much formal, planned help as possible to make things manageable.

There is a chairlift for the flight of stairs to the Sistine Chapel, and elevators to other parts of the Museums.

Main roads run in front of the museum entrance, along the Vatican wall, and in front of St. Peter's Square, but I can't say what traffic controls police might use in various situations. St. Peter's Square itself is huge, like 2 or 3 American football fields.