The jews literally bullied the governor into killing the Messiah under threat of another revolt. Had Rome made an effort to understand the religion of jews and how it was changing and the new religion of Christianity they would have picked supporting Christianity mostly because of taxes.

The jews wielded Rome against Christ, or maybe its better to say that the crucifixion was roman but the false conviction was jewish.

Rome's general rule was that upstart cults were dangerous and acceptable to purge. Rome was particular paranoid about Christianity because they believed Christians blamed them for the death of their Messiah. To the point they believed a number of fires were acts of Christian terrorism as retribution, much of this was likely political theatre.

No. They made the Roman's do it for them. It's my understanding that they basically pressured Pilate into crucifying him.