I'd guess that if you have Kodi configured to play the next episode and media is opened from a view that lists multiple episodes .. it will do what it's been told to do.
No ideas on the stop/exit although Kodi should buffer if the connectivity is poor, so it sounds more like an issue with media itself or something server-side or how the add-on works
It seems as if something is sending a skip command to the player, so if its a Movie it will just stop and exit playback, if its a TV Show it will start playing the next episode. I too expect the player simply to buffer and maybe crash out if connectivity is the issue.
What do I look for in which log file to see if this is the case?