1) How Obito awakened his MS??He didn't kill Rin.He just witnessed the scene.
2) In the last chapter I think it was clear that you don't have to kill your best friend to awaken the MS,but to go through despair by loosing a loved one.Sasuke never killed his best friend or his brother but he awakened MS.So the key thing is great emotional pain and despair.
3) Tobirama gave an explanation about the way sharingan works through an anatomical process.Special chakra that flows to the brain influence the optic nerve.Now how can be a connection through the same brain if both eyes are connected to different optic nerves??Plus he called the sharingan the mirror of feelings/the eye that reflects the feelings.So it makes more sense that the chakra that flows in the brain that makes the sharingan evolve is affected by the users feelings.
In Kakashi and Obito case both went through great pain and despair because Kakashi killed Rin and Obito witnessed it.So I believe that they both awakened their MS separately on the same time.