Best I can tell is "yes", but with a caveat. When the switch loses power that seems to reset it's clock so it won't run any rules until it resyncs with the current time - you need to be careful about that.

Once advantage this switch has over it's competition is that it returns to power in the previous state - that is, if you have a power outage your router will come back on afterwards (the competition stays off when it regains power and so does your router).

I expect you'll have to schedule the off and on separated by a minute in Kasa. If you're using my app I have a "momentary" rule and function allowing you to power cycle the switch in seconds as well as minute, and you can trigger it manually when needed too.

Yes, I have tried that, if I turn off my router, the schedule and timer still works on the device. It seems that the the plug has internal storage for its own status and the schedule. The settings are stored in the plug itself and does not rely on having a persistent internet connection to function. But plug does rely on the power to keep time so that it can act on scheduled events.