Existence of Catholic Church without hierarchy

I challenge the premise, because I think a big part of Catholic theology is God will protect his Church from this kind of event. It's not something we have to worry about... that's God's problem, to make sure it doesn't happen.

I can hear the protest now... "But what if it did?"   Stop.  God promised us it won't. If we can't trust God, what are we even doing?

That doesn't mean the Pope runs around carelessly without his Popemobile, or the Vatican should go without it guards. There's still sin in the world, and one pope, or even the entire Vatican can be replaced. But the kind total clergy wipeout described here isn't a problem for humans to worry about.

Would catholic church exist?

Surely the Catholics surviving the nature catastrophe would continue living their belief and form some organization of some kind.

If I understand your question correctly you want to know if that organization is identical to the Catholic Church today or if that organization will be some newly founded organization not being identical with the Catholic Church today.

Different people will surely answer this question differently so you have to say whose perspective you want to know.

Let's assume that the question shall be answered the same way the following question is answered:

"Is the Catholic Church in 2018 identical to the Church in 218?"

If we answer this question we only consider documents and statements which are seen as valid by the bishops in the year 2018.

When answering this question we don't ask how Christians in the year 218 would have answered this question.

If we apply the same logic to your question we would have to ask how people living after that nature catastrophe would answer your question - not how people living today answer it!

Although we cannot be sure I suspect that the faithful living after the nature catastrophe will say that the Catholic Church existing then is identical to the Catholic Church today.

In all churches with apostolic succession ...

We cannot know what the faithful after the nature catastrophe will think about how apostolic succession exactly works.

The Church militant will always exist until the end of time. Matthew 16:18:

And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it.

The papacy, and thus also the Church, will last until the end of the world, as the First Vatican Council said in Pastor Æternus (July 18, 1870), ch. II, "On the Perpetuity of the Primacy of blessed Peter in the Roman Pontiffs":

That which the Prince of Shepherds and great Shepherd of the sheep, Jesus Christ our Lord, established in the person of the blessed Apostle Peter to secure the perpetual welfare and lasting good of the Church, must, by the same institution, necessarily remain unceasingly in the Church; which, being founded upon the Rock, will stand firm to the end of the world.

Even if the vast majority of the hierarchy apostatizes, which happened during the Arian crisis,* the Church will still exist.
^{*(St. Athanasius remaining one of the few Catholics)}

St. Vincent of Lerins, in his Commonitorium book 1, chapter 2 (English translation of excerpt), foresaw the possibility of heresy taking over almost the entire Church, in which case there would remain few non-apostate, true members of the Church:

Also in the Catholic Church itself we take great care that we hold that which has been believed everywhere, always, by all. For that is truly and properly Catholic, as the very force and meaning of the word shows, which comprehends everything almost universally. And we shall observe this rule if we follow universality, antiquity, consent. We shall follow universality if we confess that one Faith to be true which the whole Church throughout the world confesses; antiquity if we in no wise depart from those interpretations which it is plain that our ancestors and fathers proclaimed; consent if in antiquity itself we eagerly follow the definitions and beliefs of all, or certainly nearly all, priests and doctors alike.

What, then, will the Catholic Christian do if any part of the Church has cut itself off from the communion of the Universal Faith? What surely but prefer the soundness of the whole body to a pestilent and corrupt member?

What if some novel contagion seek to infect the whole Church, and not merely a small portion of it? Then he will take care to cling to antiquity, which cannot now be led astray by any novel deceit.

What if in antiquity itself error be detected on the part of two or three men, or perhaps of a city, or even of a province? Then he will look to it that he prefer the decrees of an ancient General Council, if such there be, to the rashness and ignorance of a few.

But what if some error spring up concerning which nothing of this kind is to be found? Then he must take pains to find out and compare the opinions of the ancients, provided, of course, that such remained in the communion and faith of the One Catholic Church, although they lived in different times and places, conspicuous and approved teachers; and whatever he shall find to have been held, written and taught, not by one or two only, but by all equally and with one consent, openly, frequently and persistently, that he must understand is to be believed by himself also without the slightest hesitation.

Existence of Catholic Church without hierarchy?

The short answer is: The Church will continue to exist and move on with lay ministers to teach and preserve the faith.

This is what happened in Japan between 1614 and 1865.

Historically speaking it is obvious that the Catholic Church can and will continue to survive without any hierarchy whatsoever. In fact, there exists a very pointed example in the history of the Church. It is obvious that the Church will live unto the last times as our Lord has promised:

Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you always, even unto the end of the world. Amen. (Matthew 28:20).

The answer to your question is not without precedence. It has happened before. The faithful of Japan survived in great numbers for some 250 years without any Catholic priests or missionaries. That is not a little thing!

In 1614, all Catholic missionaries were expelled from Japanese soil. Fr. B. Petitjean arrived in Nagasaki, Japan in August 1864 and was able to construct a small Church. On March 17, 1865 he discovered that the Christian faith had survived in Japan for some 250 years without any missionaries (priests) to administer the sacraments!

Japan was forced to open to foreign interaction by Matthew Perry in 1853. It became possible for foreigners to live in Japan with the Harris Treaty in 1858. Many Christian clergymen were sent from Catholic, Protestant and Orthodox Churches, though proselytizing was still banned. In 1865, some of the Japanese who lived in Urakami village near Nagasaki visited the new Ōura Church which had been built by the Paris Foreign Missions Society (Missions étrangères de Paris) barely a month before. A female member of the group spoke to a French priest, Bernard Thadee Petitjean, and confessed that their families had kept the Kirishitan faith. Those Kirishitan wanted to see the statue of St. Mary with their own eyes, and to confirm that the priest was single and truly came from the pope in Rome. After this interview, many Kirishitan thronged toward Petitjean. He investigated their underground organizations and discovered that they had kept the rite of baptism and the liturgical years without European priests for nearly 250 years. Petitjean’s report surprised the Christian world; Pope Pius IX called it a miracle. - Kirishitan (Wikipeia)

In Japan, this is liturgically celebrated as the Finding of the Japanese Christians. The Finding of Christians in Japan (Inventio Chritianorum in Japoniae), or the Discovery of the Hidden Christians in Japan, is truly a most amazing feast celebrated liturgically in all of Japan. The Catholic Church in Nagasaki celebrates on March 17, the feast locally known as Shinto hakken no Seibo, which means literally "the Mother of God that rediscovered the hidden Christians." Sorry, St. Patrick's Day is not a big thing this day in Japan!

If it happened once, it could possibly happen again.

The Bells of Nagasaki (長崎の鐘 Nagasaki no Kane), August 1946 by Takashi Nagai is an excellent read on the subject.