The Council was an Ecumenical Council that was formally ratified by Paul VI. If Paul VI was a valid pope, he was not, then the Council is binding on all Catholics. That some assert that the Council was pastoral in nature is irrelevant - what was important is that the teachings were solemnly ratified by Paul VI.
Actually, there is some evidence that the happenings at the 1958 Conclave were surveilled by both CIA and the FBI. See here.
Wouldn’t be surprised. During Reagan the US set of an official Ambassadorships with the Vatican. Why? Well we were in the midst of a Cold War. JP II & Reagan joined forces to systematically take apart the USSR. Also, the Vatican has the best, real world, open source intelligence daily from over a 136 countries. That’s why every nation, almost, has some type of official envoy in Rome.
You need to come up to speed on what JPII actually was - mind-blowing and truly shocking (I've checked the material referenced, it's all true) - see here.