The red part glitch: It's difficult to imagine what could be wrong in such a simple setup. Does it still do that if you disable the skin and link the object to the red part's bone? If not: it might be a weight problem, try to assign the weights using the Skin's weight tool dialog. The rotating part: try switching to gimbal…
I checked and everything was looking fine, so I just deleted the skin and the bones and recreated them again, and now it's working! And the local coord sys too! Thanks Zhalktis! :)