I don't think I'd commented on the 10 Rings/Mandarin thing yet, but I think there are two explanations, both of which have been mentioned by other posters. Either A) Killian was behind the group the whole time or B) he just stole the image and decided to use it for Mandarin.
I'll go with C.... Marvel simply threw continuity out the window for IM3.
That being said, there really wasn't anything tying the Mandarin from IM3 to the 10 Rings group from the first movie other than fake Mandarin having 10 rings on his fingers. And the common movie-goer wouldn't notice any of that.
The more I think about it, the more I think Marvel dropped the ball with IM3. After Avengers, it would make sense to introduce alien tech (the rings) to go up against Tony-tech. A more comic inspired Mandarin would have simply been a more dangerous foe. I just couldn't buy Killian as a criminal mastermind at all.
and speaking of which, just what was Killian's motivation? Maybe I zoned out when that got explained, but to me it seemed like he went emo simply because Tony blew off their roof meeting.