Summary: Can Paul IV’s ex officio document be used to claim that Francis is not a valid Pope? No, it cannot. Is Pope Vigilius proof that Popes can be heretics? No, just the opposite.
Before becoming Roman Pontiff, Vigilius was a traitorous man, weak in faith and willing to betray the Faith in order to become Pope. And betray the Faith he did, by accepting and promoting the heresy favored by Theodora the Empress.
Saint Robert Bellarmine: “Vigilius wrote an epistle to Theodora the Empress and other heretics, whereby he confirmed their heresy and declared anathema on those who confessed that there are two natures in Christ.” 
Theodora the Empress then decreed that Vigilius was the Roman Pontiff, even though the true Pope, Saint Silverius [or Sylverius] was still in office. So Vigilius was initially an antipope, and as antipope, he asserted heresy, though mainly to Theodora and her fellow heretics. To the people of Rome, he confirmed and promoted the true Faith. Thus, it is clear that his heresy was formal, as he knew the truth and still asserted the heresy. He was also in schism, as he rejected the true Pope, Silverius.
After Vigilius continued as antipope for a brief time, the true Pope of that time, Silverius, passed away. Vigilius was then accepted as the true Roman Pontiff by the clergy and people of Rome. He then ceased to be an antipope and became the true Pope.
Then something amazing happened which serves to prove that the Roman Pontiff has the charism of truth and of never-failing faith from the prevenient grace of God (which no person can resist). From the point in time when he first became true Pope, Vigilius rejected his previous heresy and steadfastly held and taught only the true Faith.
Bellarmine: “From this time neither error nor feigning of error was discovered in Vigilius, but rather, supreme constancy in the faith even to death, as it shall appear. For he received with the pontificate the strength of faith and he was changed from a weak chaff into the most solid rock. When the Empress Theodora, having relied upon the secret letters as well as the promise of Vigilius, asked from him that he would restore the aforementioned Patriarch Anthemius, as he had promised, he wrote back that he had promised rashly and gravely sinned in that promise. Therefore, he could not, nor would, fulfill what he had promised, lest he would add sins to sins. For that reason, when the Empress became angry, he was sent into exile, and miserably tortured even to death.” 
Therefore, if ever or whenever a heretic or schismatic becomes the true Pope — either by a valid election or by acceptance as Pope by the body of Bishops — his heresy and schism, and indeed all grave errors and all failures of faith, are immediately vanquished by the prevenient grace of God in accord with the promises of Jesus in Mt 16:18 and Lk 22:32, and in accord with the dogma of Vatican I. This occurred in the case of Vigilius, without any doubt, and it may have occurred in other cases, where perhaps a grave error or failure of faith was hidden in the heart and mind, or at least unknown to the historical record.
Pope Paul IV
And, finally, this truth disproves a common misreading of Cum Ex Apostolatus Officio by Pope Paul IV. That document is often thought to assert that if a heretic becomes Pope, he is not a valid Pope but only an antipope. In fact, the document considers three types of errors: heresy, schism, and deviations. But of these three, Paul IV only attributes deviations as a possible error of a Roman Pontiff. Since the Pope is not able to fall into heresy and schism, but may in some way deviate from the Faith, said deviations must be lesser errors, not greater ones. For whoever is capable of a worse sin, is capable of a lesser sin. (Could a murderer be so trustworthy as to never tell a lie? Of course not.)
“the Roman Pontiff, who is the representative upon earth of God and our God and Lord Jesus Christ, who holds the fullness of power over peoples and kingdoms, who may judge all and be judged by none in this world, may nonetheless be contradicted if he be found to have deviated from the Faith.” 
The Pope may not be judged by anyone but God. However, the faithful may contradict the Roman Pontiff, if he is found to have deviated from the Faith, i.e. to have asserted a lesser error. But the Roman Pontiff is prevented by the prevenient grace of God from falling into greater errors, such as apostasy, heresy, or schism, or idolatry — as was later taught by Vatican I dogmatically.
And notice that the only time Paul IV considers a Pope might fall into heresy is PRIOR to his election as Pope. For at that time, he is not the Roman Pontiff, and therefore does not have the charisms of the Roman Pontiff.
“or even the Roman Pontiff, prior to his promotion or his elevation as Cardinal or Roman Pontiff, has deviated from the Catholic Faith or fallen into some heresy:” 
Heresy is not possible in anyone who is currently the Roman Pontiff. In his life, prior to becoming Pope, or afterward if he resigns, he may possibly fall into apostasy, heresy, or schism, or idolatry, or any sin or error known among fallen humanity.
And now we come to an error by Paul IV in Ex Officio — a deviation from the faith by the Roman Pontiff speaking against deviations from the faith by Roman Pontiffs.
If a man, prior to becoming Pope, is a heretic, his “promotion or elevation, even if it shall have been uncontested and by the unanimous assent of all the Cardinals, shall be null, void and worthless;” and “it shall not be possible for it to acquire validity” .
Since this claim by Paul IV is not a grave error, I am permitted by the very same document and the very same Roman Pontiff to contradict him. For the Pope is only preserved by grace from all errors when speaking infallibly, and from grave errors otherwise, and from grave failures of faith at all times. And this error by Paul IV falls outside of that divine protection and charism.
Paul IV was not here declaring any particular Pope’s reign to be null, void, and worthless. He was not a Cardinal rejecting the current Roman Pontiff, which would be a grave error. Rather, he made this assertion within a document which contains both discipline and doctrine. Most of what Cum Ex Apostolatus Officio requires is discipline, while some points of doctrine are certainly touched upon, as well. But discipline is always changeable.
Subsequently, the Popes and Councils did not continue this practice of removing persons from office or of holding their authority to be null and void, if they had ever been a heretic or schismatic at a prior time in their life. For example, a Protestant who converts to Catholicism can become a Catholic priest or a Bishop. And the recent Popes have sought to bring the heretical and schismatic group called SSPX back to the Church — and not solely as laypersons.
Can a Protestant who converts to Catholicism become a valid Roman Pontiff? Yes. And this is predicted to happen [in the near future by my eschatology] in the secrets of La Salette:
“3. ‘A protestant nation from the north will convert to the faith and, by means of this nation, the other nations will return to the faith.’
4. ‘The pope who will come after this one will not be roman.’” [The Secrets to Maximin]
If the discipline of Paul IV, on heretics who become Pope, were instead a doctrine, a Protestant who converts would be unable to become Roman Pontiff.
Now it is a dogma that heretics and schismatics are automatically excommunicated. For even without Canon law, such a sin (if it is formal) excommunicates by the very nature of the act. The heretic and the schismatic cut themselves off from the Church. Therefore, this is a doctrine of the eternal moral law, not a changeable discipline.
But when we consider the point about a Roman Pontiff being supposedly null and void if he ever were a heretic in the past, such a doctrine does not exist under the eternal moral law. Heretics cut themselves off from the Church by the very nature of their own sin (latae sententiae), and they are cut off by the law of the Church (latae sententiae) or by a judgment from Church authority (ferendae sententiae). But a repentant heretic who returns to the fold is accepted with joy as a son of the father (Lk 15). Therefore, a repentant heretic is not thereby prohibited from the offices in the Church as is stated in Cum Ex Apostolatus Officio. Such a rule is therefore of changeable discipline.
And when Paul IV’s reign ended with his death, subsequent Popes and Councils did not observe or enforce the rule. Therefore, the law on that point became null and void. For when proper authority over a law declines to observe and enforce that law, it is thereby nullified. (This is true even in the laws of secular society; a perpetually unobserved and unenforced law cannot be revived and enforced in some particular case, as this would be arbitrary and therefore unjust.)
Note well that this document by Paul IV does not teach as doctrine that any Pope who ever fell into heresy prior to becoming Pope is invalid. That would be false and disastrous, as we would never know which Popes may have been occult heretics PRIOR to becoming Pope. You can put forth a hypothetical, where such a prior heresy would be known; but just as many hypothetical cases could be proposed where the heresy would be hidden.
Moreover, the case of Vigilius proves that this discipline attempted by Paul IV is not an eternal truth. For Vigilius was a manifest formal heretic, who became the valid Roman Pontiff — as proven by his 18 years in office and by his approval of an Ecumenical Council, and by his rejection all heresy while he was Pope. Thus, the discipline of Cum Ex Apostolatus Officio, as it pertains to a Pope who was formerly a heretic, is not a dogma or doctrine, but only a discipline which never had any effect. It did not apply during the reign of Paul IV, since he was not a former heretic. And afterward, it was not continued by Popes or Councils. Indeed, Paul IV reigned during the Council of Trent, yet the Council did not approve such a proposal.
Was Francis a heretic prior to becoming Roman Pontiff? That is irrelevant. If so, all heresy and grave error was vanquished when he became the true Pope. If not, his sound faith was strengthened to become never failing when he became Pope.
Is Pope Francis the valid Roman Pontiff? He was accepted by the body of Bishops, so he must be the valid Pope. Otherwise, the Church would have lost her indefectibility, if ever the body went astray by following a false head. Can Francis lose his validity? No, for he cannot fail in faith, nor err gravely.
The case of Vigilius proves that each Roman Pontiff has a never failing faith. And the document issued by Paul IV cannot be used to claim that Francis is not a valid Roman Pontiff.
— Ronald L. Conte Jr.
 Vigilius reigned for 18 years, counting from the death of Silverius, and his acceptance as Pope by the body of Bishops proves he was, at that time, the true Pope. Vigilius also confirmed the acts of the fifth general Council (the second Council of Constantinople), also proving him to be a true Roman Pontiff.
 Bellarmine, Robert. Papal Error?: A Defense of Popes said to have Erred in Faith. Mediatrix Press. Kindle Edition.
 Cum Ex Apostolatus Officio.
Wasn’t Pope Symmachus a convert from Paganism, after the Apostolic Age and after the Roman Persecutions?
I don’t think we know the specifics of his beliefs early in his life.