1) Not sure. I heard rumors that it might be something in Act 4. You also can;t make it into a spellbook. I think.
2) I don't know about the key, but you can just lockpick it, and you will want to open it. It will drain source points, but that shouldn't be an issue.
3) That's part of Tarquins quest, the one which the game considers mandatory, and one which takes ages to finish, even though the reward is just worthless garbage. Did you find Quarra, by any chance?
One thing which is hard to remember: When in doubt,
.
5) I just sold it immediately, this is probably a quest which doesn't close properly, don't worry about it. The conclusion is "hey this necklace is stolen, wow"!
6) Sell it. That's all. (This quest seems unfinished, but apparently your choice is either "eat it immediately for phoenix dive" or sell it.)