How to place a block half-over the header?

Try moving it the CSS to the very top of the Simple CSS.
If it works then you need to check the CSS below it for any errors.

If it doesn't work, try clearing any browser or server caches.

I have another issue now:

The Button should be larger, but when I increase the image inside the container from 180 to 220, the whole header becomes bigger and shifts the content below further down.

How can I modify the button, so that it does not effect the size of the header, nor the position of the content?

Thank you in advance and kind regards,
Sascha

I have managed by changing the margins ;-)

Glad to hear that!