How was Obito able to use Kamui immediately after awakening his MS?
During the fight against the Hidden Mist ninja after Rin's death, he never teleports his entire body to the Kamui dimension. His intangibility ability is probably activated involuntarily (he never got attacked unless he purposely made himself tangible, like the fight against Minato). Kakashi doesn't have intangibility, so he wouldn't know how his MS worked until he figures out how to use Kamui.