Responses
I think you nailed it. That seems completely plausible and suits the story perfectly.
Yeah the cat somehow got killed from an accident or something. The brother didn't kill it but he thought that his brother did since he just say "the cat's not coming back". This is what happened after his brother accidentally pushed him off a cliff so he thought his brother doesn't like him so he taken the cat away from him by killing it off.
It was pretty hard to understand what was going on a lot of the time, because of the wording/translation. But I'm pretty sure that the poor kitty got hit by a car or something, and Yukito (big bro) found it. He might have tried to help it - or at least he moved it - and that's why he had blood on his hands. I think he didn't want Kuruto to see it, so he just told Kuruto that the cat was gone and wouldn't come back. But because Kururo was so young (and it seems Yukito moved out pretty soon afterwards), it makes him associate negative, scary and sad things with his bro. That's what I got, anyway - I really don't think everything would have been peachy at the end if he thought his big bro was a psycho cat murderer (￣∇￣")