Is there a possiblity to set the required user level to move a category to another category? Or is it simply possible to disable the possiblity to move a topic besides you are a admin / moderator?
Not really. Maybe read about trust levels in Discourse. Trust Level 3 is able to recategorize topics.
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Thank @MarcP. I know that TL>=3 is able to recategorize but, this is not what I want. TL4 would be OK or disable the re-categorize feature completely.
You can not do this at this moment.
Then, this is a feature request 
Or, I will work on this at the weekend.
You can change the settings to reach the Trust Level 3 so that nobody can reach it. Without the TL3 nobody will be able to recategorize the topics except the staff. Then you can assign the TL4 to the users you want to allow to move the topics between the categories.
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Yes, this can be done by disabling the trusted users can edit others site setting. See What does the “trusted users can edit others” setting do? for details about the setting.
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Small update on this one - we now have a couple of separate settings which can be group-enabled to give more control of this. 
edit all topics groups allows the added groups to be able to re-title, tag, and recategorise topics. So changing that to just eg. admins would allow you to restrict that away from TL3. 
And edit all posts groups covers the groups who have global post editing ability (default TL4 still).
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