[solbero]

When I exit kodi I come directly to the logon screen of lubuntu, is that correct?

That is the intended behaviour.

i use a externa USB-Disc for my mediastuff (movies, music, ..), but I can not access it in kodi, at "/ media" nothing is mounted (if I close kodi and login Lubuntu the hard disk is autmatically mounted and I can access the data). Can I make the disc also reachable under Kodi?

Install the package usbmount. That should solve your problem.