Mei and Akane fight about taking Keisuke away was because Mei was jealousy. Mei was worried that Keisuke liked Akane more then him because his boyfriend was worried about Akane. When Akane started to skip class and dyied his hair and Keisuke was worring over it and Mei didn't like that. Mei was just jealousy that his boyfriend was worrying more about someone else even if it was his twin brother.
As for Keisuke worrying over Akane that could be as simple as him thinking that since he is my student I should be watching over him. Or since he is Mei's twin brother he's thinking that they need to help him and deal with what ever is troubling him. Or another theory is that deep down Keisuke like Akane but didn't realize it yet, so worrying over him made his inner self feel better. You know when they draw manga and they have a character in a story that does something for the person they love but they don't realizes yet why they did it.
My theory is that Keisuke and Akane where more similar in personality then Keisuke and Mei. Mei could see that and was worried that one day Keisuke would leave him for his twin brother. You could see it in the movie that they both wanted to see. If Mei had lived I think it would have happened but would have taken years.
Anyways that is how I saw it but other people could have seen it differently then me.
The bed scene was so sweet, the rings too. I cant help but think... maybe the story was going to be different... Then the author changed their mind? why was Mei and Akane arguing about taking Keisuke away???? and Keisuke did seem really worried about Akane too????? Many unanswered questions... hhhhhmmmmmmm