I read the Emma thing as infidelity, and it was presented as such. Doesn't matter if Jean forgave him or brushed it off.
it's not. Jean confirmed by reading scott's mind that he had never been unfaithful. Emma made the offer, scott turned her down. Jean was in the wrong there.
Their relationship wasn't in a good place, but Scott was not unfaithful to her.
Cyclops was not right. Had they listened to him, Hope wouldn't have trained. Wanda would have stayed in hiding. Hope would have been consumed by the Phoenix and gone dark like Jean did and cyclops eventually would have.
Scott was correct. We see in spin off books that Hope and her "Lights" were created by the phoenix force for the express purpose of jump starting evolution. Containing the Phoenix Force is literally what she was made for. This evolutionary jumpstart had occurred elsewhere in the universe using other hosts without a problem.
There is also no reason to believe that the Phoenix is inherently an unstable or malevolent entity. The Avengers were panicked by the possibility of Dark Phoenix (who according to Byrne's explanation was never jean grey!), but completely ignored that Rachel was a phoenix host for YEARS without a problem. Quentin Quire as a stable Phoenix Avatar ALSO shows up in Battle of the Atom, perfectly fine.
Rather than listen to Scott and the X-men that the Phoenix wasn't out to destroy the planet (it wasn't) and could be controlled (it could) Tony Stark and the Avengers not only attacked the X-men to seize hope, but then **shot at** the Phoenix force and fragmented it, sending it into five DIFFERENT hosts who weren't meant for it.
Even then, the opening chapter of AvX has the Phoenix Five turning the world into a paradise and solving drought and world hunger. the Avengers go out of their way to attack them ANYWAY instead of working with them for weeks/months, which results in the phoenix five becoming progressively more unstable as the phoenix force concentrates it's power among fewer hosts, inevitably leading scott summers to go Dark Phoenix.
It was the Avengers failing to listen to the X-men and going full aggressor over a cosmic entity they did not understand that caused AvX. The Point of the series (and what formed the Unity Squad in Uncanny Avengers) is that had the Avengers had better relations with earth's mutants in the first place and listened and worked with them instead of trying to imprison them and attack the PF, AvX would never have happened.