What is the nature of the Holy Ghost's guidance of the Papacy?  There are distinctions.

The pope is infallibly prevented from making an error when pronouncing a dogma 'ex cathedra'.  In no other circuмstance is the pope infallible.  Neither in personal morals, nor in sermons, speeches and letters.

Surely the Holy Ghost offers graces of state to the pope, but where is the doctrine that the pope cannot refuse these graces?

If these recent popes were pronouncing their errors as dogma, then we would have our proof.  Note that they consistently refrain from doing this.  Only John Paul II spoke 'ex cathedra', once (and even that can be argued!), to pronounce that no woman could ever be a priest.

We have no proof.  We must behave according to the presumption that they are valid popes.  Presumption!--not certainty.

  Only the church has the authority to investigate and decide if the evidence is sufficient. 

Please provide the evidence that only the Church has the authority to investigate and decide if the evidence is sufficient.

They are removed by the law itself by tacit resignation.  That's what the law states.  I have not removed anybody.  I simply apply the law to a case. 

“Any office becomes vacant upon the fact and without any declaration by tacit resignation recognized by the law itself if a cleric: Publicly defects from the Catholic faith.”
(1917 Code of Canon Law, Canon 188.4º)

the Pope cannot destroy the Church to this extent, cannot promulgate a Mass that displeases God and harms souls, and cannot produce thoroughly corrupted "Magisterium" for 60+ years.

Repeating yourself does not make what you say any more accurate.  I asked:  where is the doctrine that says a pope cannot refuse the graces of state offered to him?  Where does it say a pope cannot cause catastrophic damage with all his non-infallible power?  I'll answer myself:  there is no such doctrine.

Obedience cannot oblige us to sin.  When the popes give bad commands, they are being schismatic, not us.  If a sergeant commands a soldier to burn down the barracks, he is being treasonous, but doesn't thereby automatically cease to be a sergeant.  The soldier who says, "Sir, no sir!" does not deny the sergeant's authority.  He recognizes the limits of the sergeant's authority.

I could say the indefectibility of the Church is more marvelously apparent in the past few years than it perhaps ever has been.  Do you realize all the battles and agony faithful priests had to go through to teach what the Church teaches in this crisis?  Rome is occupied by the largest and most ravenous pack of wolves the history of the Church has ever seen.  Not even they, with all their tricks and power, can stop the Holy Ghost from providing us with pastors who feed the flock true doctrine.  That is indefectibility!

Repeating yourself does not make what you say any more accurate.  I asked:  where is the doctrine that says a pope cannot refuse the graces of state offered to him?  Where does it say a pope cannot cause catastrophic damage with all his non-infallible power?  I'll answer myself:  there is no such doctrine.

Obedience cannot oblige us to sin.  When the popes give bad commands, they are being schismatic, not us.  If a sergeant commands a soldier to burn down the barracks, he is being treasonous, but doesn't thereby automatically cease to be a sergeant.  The soldier who says, "Sir, no sir!" does not deny the sergeant's authority.  He recognizes the limits of the sergeant's authority.

I could say the indefectibility of the Church is more marvelously apparent in the past few years than it perhaps ever has been.  Do you realize all the battles and agony faithful priests had to go through to teach what the Church teaches in this crisis?  Rome is occupied by the largest and most ravenous pack of wolves the history of the Church has ever seen.  Not even they, with all their tricks and power, can stop the Holy Ghost from providing us with pastors who feed the flock true doctrine.  That is indefectibility!

• The mere fact that a papal teaching is not protected from all error does not necessarily mean that it could be heretical
• Catholics have an obligation to assent to all papal teaching, infallible or not
• The Pope’s teaching cannot be heretical, nor can it contain any other harmful error
• We Sedevacantists do not reject submission to the Pope, we reject someone’s claim to being the Pope

Actually only those are to be included as members of the Church who have been baptized and profess the true faith, and who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed. …
Nor must one imagine that the Body of the Church, just because it bears the name of Christ, is made up during the days of its earthly pilgrimage only of members conspicuous for their holiness, or that it consists only of those whom God has predestined to eternal happiness. it is owing to the Savior’s infinite mercy that place is allowed in His Mystical Body here below for those whom, of old, He did not exclude from the banquet. For not every sin, however grave it may be, is such as of its own nature to sever a man from the Body of the Church, as does schism or heresy or apostasy.
(Pope Pius XII, Encyclical Mystici Corporis, nn. 22-23; underlining added.)

We likewise define that the holy Apostolic See, and the Roman Pontiff, hold the primacy throughout the entire world; and that the Roman Pontiff himself is the successor of blessed Peter, the chief of the Apostles, and the true vicar of Christ, and that he is the head of the entire Church, and the father and teacher of all Christians; and that full power was given to him in blessed Peter by our Lord Jesus Christ, to feed, rule, and govern the universal Church; just as is contained in the acts of the ecuмenical Councils and in the sacred canons.
(Pope Eugene IV, Council of Florence, Bull Laetentur Coeli; Denz. 694.)
But the supreme teacher in the Church is the Roman Pontiff. Union of minds, therefore, requires, together with a perfect accord in the one faith, complete submission and obedience of will to the Church and to the Roman Pontiff, as to God Himself. This obedience should, however, be perfect, because it is enjoined by faith itself, and has this in common with faith, that it cannot be given in shreds; nay, were it not absolute and perfect in every particular, it might wear the name of obedience, but its essence would disappear….
In defining the limits of the obedience owed to the pastors of souls, but most of all to the authority of the Roman Pontiff, it must not be supposed that it is only to be yielded in relation to dogmas of which the obstinate denial cannot be disjoined from the crime of heresy. Nay, further, it is not enough sincerely and firmly to assent to doctrines which, though not defined by any solemn pronouncement of the Church, are by her proposed to belief, as divinely revealed, in her common and universal teaching, and which the [First] Vatican Council declared are to be believed “with Catholic and divine faith.” But this likewise must be reckoned amongst the duties of Christians, that they allow themselves to be ruled and directed by the authority and leadership of bishops, and, above all, of the Apostolic See.
(Pope Leo XIII, Encyclical Sapientiae Christianae, nn. 22, 24)
All know to whom the teaching authority of the Church has been given by God: he, then, possesses a perfect right to speak as he wishes and when he thinks it opportune. The duty of others is to hearken to him reverently when he speaks and to carry out what he says.
(Pope Benedict XV, Encyclical Ad Beatissimi Apostolorum, n. 22)
…[T]his sacred Office of Teacher in matters of faith and morals must be the proximate and universal criterion of truth for all theologians, since to it has been entrusted by Christ Our Lord the whole deposit of faith — Sacred Scripture and divine Tradition — to be preserved, guarded and interpreted….
Nor must it be thought that what is expounded in Encyclical Letters does not of itself demand consent, since in writing such Letters the Popes do not exercise the supreme power of their Teaching Authority. For these matters are taught with the ordinary teaching authority, of which it is true to say: “He who heareth you, heareth me” [Lk 10:16]; and generally what is expounded and inculcated in Encyclical Letters already for other reasons appertains to Catholic doctrine.
(Pope Pius XII, Encyclical Humani Generis, nn. 18, 20)
Therefore, when it is a question of instructions and propositions which the properly constituted shepherds (i.e. the Roman Pontiff for the whole Church, and the Bishops for the faithful entrusted to them) publish on matters within the natural law, the faithful must not invoke that saying (which is wont to be employed with respect to opinions of individuals): “the strength of the authority is no more than the strength of the arguments.” Hence, even though to someone, certain declarations of the Church may not seem proved by the arguments put forward, his obligation to obey still remains.
(Pope Pius XII, Allocution Magnificate Dominum)

Despite the divergent views about the existence of the infallible pontifical teaching in the encyclical letters, there is one point on which all theologians are manifestly in agreement. They are all convinced that all Catholics are bound in conscience to give a definite internal religious assent to those doctrines which the Holy Father teaches when he speaks to the universal Church of God on earth without employing his God-given charism of infallibility. Thus, prescinding from the question as to whether any individual encyclical or group of encyclicals may be said to contain specifically infallible teaching, all theologians are in agreement that this religious assent must be accorded the teachings which the Sovereign Pontiff includes in these docuмents. This assent is due, as Lercher has noted, until the Church might choose to modify the teaching previously presented or until proportionately serious reasons for abandoning the non-infallible teaching contained in a pontifical docuмent might appear. It goes without saying that any reason which would justify the relinquishing of a position taken in a pontifical statement would have to be very serious indeed.
(Rev. Joseph Clifford Fenton, “The Doctrinal Authority of Papal Encyclicals, Part I”, American Ecclesiastical Review CXXI [August, 1949], p. 144)

It might be definitely understood, however, that the Catholic’s duty to accept the teachings conveyed in the encyclicals even when the Holy Father does not propose such teachings as a part of his infallible magisterium is not based merely upon the dicta of the theologians. The authority which imposes this obligation is that of the Roman Pontiff himself. To the Holy Father’s responsibility of caring for the sheep of Christ’s fold, there corresponds, on the part of the Church’s membership, the basic obligation of following his directions, in doctrinal as well as disciplinary matters. In this field, God has given the Holy Father a kind of infallibility distinct from the charism of doctrinal infallibility in the strict sense. He has so constructed and ordered the Church that those who follow the directives given to the entire kingdom of God on earth will never be brought into the position of ruining themselves spiritually through this obedience. Our Lord dwells within His Church in such a way that those who obey disciplinary and doctrinal directives of this society can never find themselves displeasing God through their adherence to the teachings and the commands given to the universal Church militant. Hence there can be no valid reason to discountenance even the non-infallible teaching authority of Christ’s vicar on earth.
(Fenton, “The Doctrinal Authority of Papal Encyclicals, Part I”, pp. 144-145; underlining added.)

I am the good shepherd. The good shepherd giveth his life for his sheep. But the hireling, and he that is not the shepherd, whose own the sheep are not, seeth the wolf coming, and leaveth the sheep, and flieth: and the wolf catcheth, and scattereth the sheep: And the hireling flieth, because he is a hireling: and he hath no care for the sheep. I am the good shepherd; and I know mine, and mine know me.
(John 10:11-14,4-5)

• The “But we’ve had Bad Popes before” Objection
• St. Robert Bellarmine defends Popes accused of Error
• Do Catholics have to Assent to Non-Infallible Church Teaching?

During the lapse of centuries, the mystical Spouse of Christ has never been contaminated, nor can she ever in the future be contaminated, as Cyprian bears witness: “The Bride of Christ cannot be made false to her Spouse: she is incorrupt and modest. She knows but one dwelling, she guards the sanctity of the nuptial chamber chastely and modestly” [De Cath. Ecclesiae unitate, 6].
(Pope Pius XI, Encyclical Mortalium Animos, n. 10)

Let no man deceive you by any means, for unless there come a revolt first, and the man of sin be revealed, the son of perdition Who opposeth, and is lifted up above all that is called God, or that is worshipped, so that he sitteth in the temple of God, shewing himself as if he were God. Remember you not, that when I was yet with you, I told you these things? And now you know what withholdeth, that he may be revealed in his time. For the mystery of iniquity already worketh; only that he who now holdeth, do hold, until he be taken out of the way. And then that wicked one shall be revealed whom the Lord Jesus shall kill with the spirit of his mouth; and shall destroy with the brightness of his coming, him, whose coming is according to the working of Satan, in all power, and signs, and lying wonders, and in all seduction of iniquity to them that perish; because they receive not the love of the truth, that they might be saved. Therefore God shall send them the operation of error, to believe lying: that all may be judged who have not believed the truth, but have consented to iniquity.
(2 Thessalonians 2:3-11)

• The Pope and the Antichrist: The Great Apostasy foretold