Yes, once your router comes back online, your thermostat will connect to the router.

If your Wi-Fi connection is restored and your router and its password are the same, your thermostat will automatically reconnect.

Yes, the thermostat will operate the heating and/or cooling system with or without Wi-Fi. Losing your Wi-Fi connection will not prevent the thermostat from running schedules – it will just prevent control over the thermostat from a mobile device until the connection is restored.