Correct, it sounds like you will have to control the light on/off with the cain on the fan. Unless you want to go to the expense of running more wiring and another switch for the light to run...

It does sound faulty to me. You could try that extender just for testing though.

Try resetting one to factory settings and setting up again. That helped when i had issues on one of my plugs

Oh also have you tried another device like a tablet or old phone?

Try logging out of the app and logging back in. If that doesn't work try uninstalling and reinstalling the Kasa app.

Maybe try doing a facotry reset on the device as well and run set up again.