If I lose internet, will my programmed schedules still occur?

Yes. Schedules are stored in the Decora Smart Wi-Fi devices.

When a Decora Smart Wi-Fi device connects to the My Leviton Cloud Service it receives current time, the location sunrise/sunset information and schedules.  As long as power remains the schedules programmed via the My Leviton App will continue to run even if the Internet service is lost.