@TobiasP,
I found some time in between some other actions/tasks.
You should have a green button under "Tools & Settings > General Settings > PHP Settings (click)" for both PHP 5.6.20 and PHP 7.0.5.
If you have a grey button, just click on it to activate PHP FPM. Afterwards, go to a domain and select PHP settings: you should be able to change to the "FPM served by Apache" setting.
If that does not work out, just go to the command line and check the status, run: service plesk-php56-fpm status
If that gives you some error notification, then we have to dig deeper.
For that reason, can you also have a look at the installed packages: do you have psa-phpfpm-configurator installed?
Regards....