Is it possible to set a different frontend- and backend-language?

You can set the site language to whatever you wish in the Settings > General > Site Language.

After that, you can head over to your profile and choose a language for the back-end. To do so, head over to Users > Your Profile > Language and change it to English.

However, there is an issue with Admin-Ajax requests. If you set the back-end language to English, the Ajax requests will be in English too.