About the first one, sorry but what do you mean by lying? Though yes, it stays in the inventory as the quest isn't actually completed in the demo. The Completed Quest note is only so the demo players won't waste their time looking for Thane.
About the second, it's because you gave it right away I didn't have the quest listed in the first place since there was no need for the memo. Sorry if these cause any confusion in the demo, I'll polish those in the full game, thx for the info! ^ ^