Which is odd, as I've just tested it myself with my own server and had no issues...I did the default load, yes.
I tried using the commands to complete quests 35 and 2, but the result is the same as when I try completing quest 13: it says "Completed quest etc" in the chat, but in the Q.U.E.S.T. Tablet v2, everything is still as in my screenshots, ie. not completed. I don't understand why I can't mark them as complete when I'm doing bq_admin edit. I even tried removing the links from 13 to 35 and even deleting 13, but to no avail. That's very weird, even more because I have no problem whatsoever when playing in a local game.
Are you using a different server thing, like Thermos or Cauldron or something (not sure what is in use nowadays)