I have two questions that I can not answer to myself:

How can I change the size of my window, if I do not know the exact size of the phone screen? I.e. my aim is to fit all screen sizes.
Is there any difference between clicking with mouse and touching with fingers in the code? If I write code for clicking, will it work with touch?
On mobile, your app should automatically fill the phone screen. You don't need to worry about it. On desktop, you can use the --size=WxH option to test a specific screen size, or use the screen module (-m screen:nexus7 for example - run kivy with -m screen to see the available options).
No. All mouse/touchscreen interactions are considered touches in Kivy. So using on_touch_down/on_touch_move/on_touch_up will work regardless of the input device. The only difference is that with touchscreen you could have multi-touch - but if you write your app assuming single-touch it will work the same on both mobile and desktop.
Kivy: how to change window size properties and the difference between click and touch