Plex Media Server doesn´t start after installation

There is no such option. You cannot just open a media file “with Plex” as if Plex was a normal media player. It is not.

Start "Plex Media Server" using the regular Windows Start menu.
Then open the Windows task tray area and look for the Plex logo.

Once you see this, double-click on the Plex logo.
Then continue here: New Claiming Requirement for Windows