Correct, the Catholic Church was completely uninvolved. Because Salem was Puritan, not Catholic. And yes, the Puritan church was very very much involved. Reverend Parris is the person who instigated the paranoia and made the first accusations, (technically, his children, but probably him) and he played an active role throughout, and he was not the only reverend involved.
It's true, it was the court who actually convicted people, but they were still all Puritan.
I'm not sure about Europe, but I think those were Catholics. Maybe not, though.
And no one actually burned in Salem, they hung people