How to switch from demo to full version after purchase?

You can delete the activation as demo in the preferences and restart the software. Now you can enter your serial number for the full version.

Can I move my eventset with music to another computer or os or take it with me with an usb stick?

Yes. All loaded audio files in the directory tree of the eventset file can be copied or moved with the eventset file an will be found on another system or computer. In this case pathes are saved as relative paths.
Paths of loaded files from another locations are saved with absolute paths and so will not be found on another computer or operating system.