Ok.... we did the next step and have been able to solve at least a part of the problem.
We assigned the german menu items to the german language (besides "Home" because that has to stay on "All languages"), copied them and assigned these new items to the english language - and translated them.
It seems to work for now. But if you change languages the site always jumps back to the homesite and doesn't stay at the menu item you have chosen before.
For example you visited the item "Manufacture" and change the language - you will always get back to "Home"
Maybe we can also solve this little problem