Yes, that is possible. You can connect to the Web Gui, in the advanced settings there is the item "Circuit diagram". Here you can define times to switch the Extender on and off. This may depend on the HW version, I have HW version 7.

Only the LEDs can be controlled in terms of time.

If you set it in the Fritzbox, the TP Link does the same.

No, nothing can be configured to be plugged in or out. But it works super well, I have 3 pieces distributed around the house.