Tap the + button on the main screen. Enter the bill name, amount, due date, and how often it recurs. That's it.
BillPing uses iOS local notifications. You can choose to be reminded on the due date, 1 day before, or 3 days before. Make sure notifications are enabled for BillPing in your iPhone Settings.
BillPing stores all data locally on your device. Your data is included in iCloud backups if you have iCloud Backup enabled on your iPhone.
Not currently, but this is on our roadmap.
Go to iPhone Settings → Notifications → BillPing and make sure notifications are allowed. Also check that Do Not Disturb or Focus modes aren't blocking them.