It wasn't everyone hating each other so much as Shatner vs. everyone else. The only one who says anything nice about Shatner is Nimoy.
I sort of feel for Shatner. I saw a BBC interview the other day where he mentioned (probably not for the first time) how he first found out what they thought of him. It was years after TOS, and he was going to write a book, so he decided to visit the old cast to reconnect and pick up some anecdotes. He spent a day hanging out at George Takei's house, and he says he had a really great time. Next day he goes to Nichelle Nichols' house, and... same thing. But then when he's about to leave, she hesitantly asks him "Don't you want to know... why we all hated you?" and he was kind of stunned. So he sat back down and she explained it all to him, and it took him a long time to absorb it all.
Must have also been a pretty weird feeling for George Takei, to have Shatner just drop by to hang out.
Seems to me there are a lot of people who act like assholes when they're at work, and it's really easy to hate on "that asshole from work", knowing you're never going to have to do more than work with them, but then the Trek actors and Shatner got put in a weird situation by getting lumped together as some kind of default friends.
Also, regarding Shatner and Doohan shortly before Doohan's passing, Shatner apparently met with Doohan privately before the event, but Shatner didn't have the emotional strength to do it onstage, so he left the con early, and Doohan understood. To the public, it looked like Shatner was snubbing Doohan and ramping up the feud, but it was actually the opposite. Shatner has also said that it was Doohan-specifically who first convinced him to give conventions a chance, back when he was refusing to go to them out of ignorant pride.