Burn is estimated from time, heart rate, distance, steps and your weight, calibrated to your Apple Watch history (±20%). Steps already inside your logged walks and Stairmaster aren't counted twice. MyFitnessPal logs tend to run about 20% under, so treat the daily number as a guide and the weekly weight trend as the truth.
Burn/day is the average estimated training burn on days you trained.
Everything stays on your device once downloaded. Keep a backup somewhere outside AZM, such as iCloud Drive.