The first four levels are free. Level 5 onward requires the one-time $2.99 Full Game Unlock.
Tap Restore in the main menu. Restoring requires a network connection so the App Store can confirm your purchase.
No. Progress is stored only on the device. There is no iCloud sync. Deleting the app deletes your save.
Yes, from within the app. Be careful: the reset wipes campaign progress irreversibly and cannot be undone.