Do you pay her if you cancel last minute: Yes. She is available and set to work her scheduled hours in the contract regardless of illness on your end causing the cancellation.
Sick Time for her: I lean toward no, but I'm full time, almost 365 days a year, and I'm allowed 3 paid sick days per year. A little ridiculous, but I've also rarely used them in the years I've been with my family so it has never been an issue.
Carseat: Ask if she is willing to leave it installed in the car, if not, teaching her to properly install it and just checking it on the day she has your LO seems like less of a pain than you installing it every single day. When I still used my own vehicle I had my own seats, took them to the fire department to check proper installation, and rarely, if ever, removed them.