What are the circumstances of what Obito the Sharingan used against Konan and the Kakashi's Sharingan?
If you have seen the fight between Tobi (Obito) and Konan, he actually used his left eye which he had given to Kakashi. So the Sharingan he used for Izanagi was of the other (Uchiha). 



Also, Obito gave Kakashi his Sharingan (the right one) and Kakashi already has the left Sharingan, so he has both and talking about Mangekyo, Kakashi has already awakened that power.