These, in very concise form, are the same responses many of us have been making to these objections here at CI.
Really the most serious objections are
1) the perpetual successors
and
5) universal acceptance
Father Cekada did not spend enough time on #1 to really refute the argument. What does it mean for the "power" of the papacy to be perpetual? It's obvious that the absence of an actual Pope does not cause the power to cease. Are we talking about juridical continuity? What do we mean here? He should elaborate on that.
With regard to #5, the reason that universal acceptance is considered the criterion for establishing the dogmatic fact of the papacy is precisely because theologians consider this universal acceptance (by the episcopate) to be infallible, since anything else would lead to a defection of the Church. Yet these same bishops can fail by universally teaching the errors of Vatican II? That doesn't make sense. If they can fail by promulgating Vatican II and universally accepting and promulgating the New Mass, then why can't they have failed in acknowledging a false Pope? Such acknowledgement cannot con-validate a faulty election either.
What's interesting, though, is that the infallibility of the episcopate has always been tied to the papacy. If the bishops were to hold an Ecuмenical Council but in the end the Pope were to reject it as a "Robber Council", then it would not have any infallibility or authority on its own.