1. John XXIII: As I understand this is where most of the controversy lies. Was John XXIII elected an anti pope or did he fall sometime during his pontificate? Perhaps when he signed off on the first heretical council document?
There is not too much controversy with John XXIII. He did things which cast grave suspicion on his orthodoxy, but there is no smoking gun, with which someone can say with absolute certainty, "he defected, he's a public heretic." I think most Sedevacantists just say, "let's retreat to the last certainly orthodox pope."
This question of John XXIII will remain a gray area until the Church decides it, in by opinion. I do not believe that there is strong enough evidence to say with certainty that he defected and became a public heretic.
2. Paul VI: In light of what happened with John XXIII was the conclave that elected him valid? Was he at anytime a valid pope?
On December 7, 1965, Paul VI publicly professed heresy and grave error to the universal Church in Council. There can be no doubt that he was not a pope on that day. The argument against him from the time of his election, June 21, 1963, was that since he showed clearly only about two and a half years later that he was a heretic, and there was no known change in his thinking, that one could form moral certainty that he was a heretic right from the start of his "pontificate."
I have no doubt that when the Church reforms Paul VI's "pontificate" will be declared null and that he was a heretic, a destroyer and an antipope.
3. John Paul I: Not much is said about him because his pontificate (or lack thereof) was so short, but again, was the conclave invalid or something or did he do something that confirmed himself as a public heretic? Could he have been a valid pope?
He will also remain a mystery. I do not know of him professing public heresy. He didn't do anything of any significance during his time, so it is not really a pressing question. The Church will judge him in due time, but I see no reason that anyone must concern themselves with him.
I guess the overriding question here is what are the Catholic principals that govern what has happened to the papacy during and after the Vatican II council?
The governing principle is found in the answer to these questions: Is it a public and provable fact that these men (Paul VI, John Paul II, Benedict XVI and Francis) have defected from the Faith? Is there strong enough evidence against them which can lead a Catholic to moral certainty that they are not popes? Is it a fact that these Popes have done things that Popes cannot do, such as promulgate a sacramental rite that leads to impiety, or bind the Church to heretical communion rules.
If you answer "yes" to those questions,then you have the basis to withdraw from these men and sever communion with them. Catholics are strictly forbidden to remain in communion with heretics. At the same time, however, you are bound to remain in communion with all Catholics who have not formed this judgment and remain under the undeclared heretical antipopes.