Question about Xiaomi X20+

So I can only start the robot without wi-fi using the button, not using the pre-configured schedule with my settings?

I have not yet tried it out

I understand that if I want to control it remotely it has to be connected to the internet. But once everything is set up, doesn't it control itself even without the Internet?

Yes the Wifi connection is indispensable otherwise you cannot control the robot and its functions via app