This looks ok so far, but please select „All“ in Remote Control Diagnostics and copy and paste the full output so I get a complete picture.

Regarding Plex itself, there is a possibility that it is put to sleep by App Nap. Since another user reported similar issues with XBMC, from which Plex was forked, this is well possible.

Please see my remarks on how to use Activity Monitor in my reply there and do essentially the same, just with Plex instead of XBMC ;-)

http://www.iospirit.com/support/forums/remotebuddy/singlethread/16565/

Thanks in advance.

Best regards, 
Felix Schwarz