The performance gain between PHP 5.6 and 7.x is huge so yes, it will help.
1 minutes to open http://tsv-schnait.net/aktuelles/akt-infos.html sounds strange, it's not really what I call a lots of nodes But maybe the nesting as an impact. @dimaip can maybe help us here with more details ?
The new UI perform also a lots better on big pages. So if you update the Neos 3.2, you can try the React based UI (in paralel with the Ember based UI). It can be really intersting for us to have your feedback about the performance of the new UI. Check the readme for more informations about how to enable the new UI: https://github.com/neos/neos-ui#installation-and-usage
Hard to say without a full access to your setup and Fusion configuration. An audit can certainly improve performance, but hard to say how much.
From my side, let's try:
- Upgrade to the latest Neos version
- Update PHP to 7.x
- Try the react based UI
Try to test the performance between each step, and if you can update the post with the details, it can be useful for the community.