Yes you can do the helm quest again it just takes some time.To get the mystery weapon quest you gotta be a certain level (I forgot what it is)and you gotta talk to the Mage in the other museum room but you gotta buy the recipe for the weapon first using platinum.Then just keep talking to her.For the gem membership quest I suggest you go to the victory lap and kill the bosses there be causes that's how I completed the quest.I hope this helped.