Did ghost and Cairn hate eachother?
Ghost never did any thing wrong. In fact, Ghost was as kind as they could be.
Ghost talks about cairn in a pretty loving way, and ghost never seems to keep choices from cairn
Ghost also refers to cairn as "the kid" but that makes sense considering cairn didn't have a name then. But Ghost also always tries to give Cairn choice. Please note: even with all their dangerous movements, Ghost never tried to control Cairn
Ghost never questions Cairns brash movements and even seem to sympathise with him at some points.
Ghost never really blames Cairn for anything (other than lapis not coming back, but it should be noted he blames himself too) They just call him Annoying, Harsh, and rude in their final moments.
They don't hate eachother and I don't think they blame eachother either. They're just tired of having to share a consciousness. Both have such apposing attitudes, who could blame them?
Cairn not getting a response and going from that tired lazy eyed look to wide eyed realisation... Even though cairn says it now, I really don't think they hated eachother. I really don't think they wanted the other gone/shattered until Acuntmea told cairn that ghost stole all choice from them, when that clearly wasn't the case.
Both Cairn and Ghost tried to make it work, but Acuntmea, that bastard, took advantage of Cairns insecurities of choice and freewill and THAT is why Cairn wanted Ghost completely gone.
Cairn was convinced by Acuntmea that their relationship with ghost was controlling when ghost really did try to give Cairn just as much choice as possible. Cairn was just scared that they were still second pilot in their life. It should be noted that I don't think Cairn at any point in the manga has really bad mouthed Ghost.