I could solve two of the problems by myself. Regarding to place the logo, I used “Custom CSS” and applied some left margin for the logo. That worked well. Regarding the space I had to select the checkbox for disabling to set the content title.
Regarding the seaarch field. I already tried your solution, but you can see on the picture below, that if you want to search something, the text overlaps the menu. That doesn’t look very nice. Is there another solution?