Do you recap stairs, or does the whole staircase come out?
Recapping is usually enough. Hardwood treads and risers go over the existing structure, with new railings, spindles and posts if you want them, which is faster and cheaper than a rebuild and leaves the surrounding drywall alone. A full rebuild is for when the structure itself is wrong, and Alexander will tell you plainly which one your stairs need.
Can you work around us while we are living in the house?
Yes, that is the normal case here. Work is staged room by room so you keep a path through the house, dust control goes up before sanding starts, and the site is tidied at the end of every day. The timeline in your quote assumes the house is occupied.
Do you cover Cambridge and Elmira as well as Kitchener-Waterloo?
Yes, along with the townships around them, all from the Kitchener office. If you are just outside that ring, call and ask before assuming the answer is no.