Gai because right now he is not f*cking the shit up for putting the world in eternal genjutsu; if he would have to choose he would choose Gai.
And "eternal rivals" are like best friends/bros.
Both. Gai for sure, but despite the history with Obito, I believe that he cares deeply for him, like Naruto cares deeply for Sasuke when he was 'bad'. Despite that Kakashi cares for Obito, he would protect his other beloved ones.
I think he cares equal for them lets not forget his whole ninja way was cause of obito yet and still Gai is his eternal rival. Side note have you notice all the strong leaf ninja has had a friend that ended up on the bad side hashi the 3rd jariya kakashi and naruto who also I believe led them stronger.