What does the tablet icon in the sidebar mean, and why is it red or green?

That's your Supernote device connection. The tablet icon is RED with "Device not connected" when the app can't reach your Supernote over Wi-Fi, and GREEN (showing the device's name, or "Supernote connected") once it's reachable on the same network. Click it to open the connection panel to connect or check status. Your tasks still sync through the Supernote cloud even when the device itself isn't directly connected.