Menu > Settings > Phone > Alerts > Toggle to Off. If the alerts come back, then you will need to look under device settings in Garmin Connect as they will eventually overwrite what's been set on the watch.
None. The Alert On/Off used to be a little more logical on the Fenix 3 when the Phone area of the settings was called Bluetooth; so it was an alert to say Bluetooth was connected or not. But as the Fenix 5 series can connect to more than just a phone via BLE, that area of the Menu had to be renamed to Phone and naturally one could easily think Alert means something else in that context.