Schedules and timers are local to the device, so if your internet goes out it won't impact existing settings on the device. In addition to the cloud functionality, they also have a local protocol that works if the cloud service is unavailable but your local WiFi is still functioning. The gotcha is that they just store state change events in the schedule, so it knows to turn off the socket at 8pm, and turn it on at 8am, but doesn't understand that it should be off between 8pm and 8am, so if your power is out when a timer or schedule should do something, that something won't happen, and the switch will come back up in the state it was when the power was lost.
What happens if you have a doser come on for 5 minutes and the power goes out. When the power comes back on does the doser keep running nonstop? This is very important to know because if it does I won't be hooking up dosers this way.