Email support@turnorest.com. Tell us what your rota looks like — it is the single most useful thing you can send, and it is what changes the app most.
Settings › Draw my own pattern lets you build any cycle, day by day, of any length. If it still does not fit, write to us: patterns that do not fit are the feedback we most want.
If iCloud was on, it comes back on its own once you sign in. If it was not, open Files › Turno › Backups on the old phone, send yourself the most recent JSON file, and use Settings › Import from file.
Settings › Apple ID › Subscriptions on your iPhone. The lifetime purchase does not renew and has nothing to cancel.
Refunds are handled by Apple, at reportaproblem.apple.com.