No, the Apostles didn't pick St. Peter.  St. Peter was picked directly by Our Lord, as per the Gospel of St. Matthew 16:18 - 19.

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.  19 And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose upon earth, it shall be loosed also in heaven.

Now, as to how the successors were chosen, that was determined by St. Peter himself, and at one point he designated that the priests of his diocese (clergy of Rome) would select the next Pope.  St. Peter could have just designated his successor, etc.  But since the Pope is the head of the Church, he can decide upon however it should be done.

Many regions of the Church later adopted the mechanism to have the clergy (and sometimes also acclamation of the faithful) to select their next bishop.

One might infer that this custom was of divine institution, but we can't be sure.

St. Robert Bellarmine, however, addresses a hypothetical scenario where all the Cardinals are killed (which these days could actually be a possibility, where someone could simply bomb the Conclave) ... and he states that the Cardinalate is not of divine institution and that the Church would always have a way to choose a pope.  Were the Cardinals all killed, there could be an Imperfect Council, a Council of the world's bishops, etc.  Cardinals, BTW, are simply stand-ins for the "clergy of Rome", and that's why they're all assigned "titular" churches.  I believe the clergy of Rome were replaced by Cardinals once the Church came out of persecution and travel was easier, etc. ... and since the Bishop of Rome ruled not only that diocese but the entire world, it made sense to recruit electors from around the broader Church and not limit them to just the clergy of Rome.

So, the election process differs from a "democratic" one because the election does not in fact confer the papal authority on the candidate, since authority does not come from below but from above, i.e. from God.  So the Church just says, "we pick this guy" and God respects that decision by conferring papal authority on the elected.  That's actually the root of the formal/material distinction in terms of papal election (that principle is very solid and not made up as some SVs like to pretend).  In the case of a bishop, the clergy / faithful could designate a bishop (aka nominate) but only if the Pope approves and invests him with authority does that elected receive episcopal authority.

The chances of a traditional pope being elected nowadays is zero, no matter how many prayers are said.

It will be no different to every pope elected since John XXIII.

The media is awash with the need for a pope who will continue with the 'progressive' Church of Pope Francis.

Just look at the end he got in both Church and media and you would think the world has gone 'Catholic.'

In no time we will have another saint.The only hope is divine miraculous chastisement, like the roof falling

down as they pick the next modernist.  How in God's name did He allow His Church to become 'the seat of the

Antichrist' as our Lady of La Salette is said to have prophesied. .

OK?  So what's the problem?  This is the nature of this chastisement, a sifting of the faith, the faithful from the unfaithful (those who just clung to the Church for cultural reasons or on automatic pilot, and not because they actually discerned the true faith and the true Church).  So while strengthening the faith of those who pass the test, it's weeding out those who didn't really have it but were just glommed onto the Body of Christ like some kind of accretion.  It's that end-times sifting Our Lord predicted.

Those of us who have the Catholic faith are in no ways troubled by the natural perspective on the situation.  God is in complete control and ending the Crisis would be for Him mere childsplay.  Do we have such little faith?  Yes, this is a test of faith.  In His time, God will remove the wicked infiltrators and restore the Church.  Who cares about what this apostate Nonclave is going to do?

A sedevacantist website called the WM Review did a study of about half a dozen theologians who taught what should be done if there were no college of cardinals, or if they were somehow unable to elect a pope, such as by being locked in dungeons or something.

There are basically two opinions of theologians, one being that the clergy of the diocese of Rome would have the right to elect the pope, and the other is that the bishops of the world could get together in council and elect a pope. The latter opinion seemed to have a bit more support, and the arguments in favor of it struck me as more convincing.

Here's an example of what St. Robert Bellarmine said on this subject:

Book I, Chapter 10
Eighth Proposition

Quote

If there were no papal constitution on the election of the Supreme Pontiff; or if by some chance all the electors designated by law, that is, all the Cardinals, perished simultaneously, the right of election would pertain to the neighboring bishops and the Roman clergy, but with some dependence on a general council of bishops.

In this proposition, there does not appear to be universal agreement. Some think that, exclusive of positive law, the right of election would devolve on a Council of Bishops, as Cajetan, tract. De Potestate Papae & Concilii, cap. 13 & 21 & Francis Victoria, relect. 2. quest. 2. De potestate Ecclesiae.
Others, as Sylvester relates s.v. Excommunicatio, 9. sec. 3, teach that in that case the right of election pertains to the Roman clergy.
But these two opinions can be reconciled. Without a doubt, the primary authority of election in that case pertains to a Council of Bishops; since, when the Pontiff dies, there is no higher authority in the Church than that of a general Council: and if the Pontiff were not the Bishop of Rome, or any other particular place, but only the general Pastor of the whole Church, it would pertain to the Bishops either to elect his successor, or to designate the electors: nevertheless, after the Pontificate of the world was joined to the bishopric of the City [posteaquam unitus est Pontificatus orbis Episcopatui Urbis], the immediate authority of electing in that case would have to be permitted by the bishops of the whole world to the neighboring bishops, and to the clerics of the Roman Church, which is proved in two ways.
First, because the right of election was transferred from all the neighboring bishops and the Roman clergy to the Cardinals, who are a certain part of the bishops and clergy of the Roman Church; therefore, when the Cardinals are lacking, the right of election ought to return to all the bishops and clergy of the Roman Church.
Second, because this is a most ancient custom (as we showed above from Cyprian), that the neighboring bishops, in the presence of the clergy, should elect both the Bishop of Rome and others also. And it is unheard of that the Bishops or Archbishops of the whole world should meet for the election of the Supreme Pontiff, except in a case where it is doubtful who should be the legitimate electors. For this doubt ought to be resolved by a general Council, as was done in the Council of Constance.

If I was Peter I would have laid down the rule that only a bishop or cardinal that professed the Faith as it was laid out over the first centuries could become a pope. That would have prevented the chaos within the Church today.

.

Our Lord laid down this rule, and it is a constant tradition in the Church that a heretic cannot hold any office in the Church. St. Robert Bellarmine says this position is the constant, unanimous position of the Fathers. It is enshrined in canon law as well, and has always been taught by theologians. Canon 188.4 of the 1917 code says that a cleric loses his office automatically, without any operation of law, in a manner equivalent to tacit resignation if he publicly departs from the Catholic Faith.

There are basically two opinions of theologians, one being that the clergy of the diocese of Rome would have the right to elect the pope, and the other is that the bishops of the world could get together in council and elect a pope. The latter opinion seemed to have a bit more support, and the arguments in favor of it struck me as more convincing.

Here's an example of what St. Robert Bellarmine said on this subject:
In this proposition, there does not appear to be universal agreement. Some think that, exclusive of positive law, the right of election would devolve on a Council of Bishops, as Cajetan, tract. De Potestate Papae & Concilii, cap. 13 & 21 & Francis Victoria, relect. 2. quest. 2. De potestate Ecclesiae.
Others, as Sylvester relates s.v. Excommunicatio, 9. sec. 3, teach that in that case the right of election pertains to the Roman clergy.
But these two opinions can be reconciled. Without a doubt, the primary authority of election in that case pertains to a Council of Bishops; since, when the Pontiff dies, there is no higher authority in the Church than that of a general Council: and if the Pontiff were not the Bishop of Rome, or any other particular place, but only the general Pastor of the whole Church, it would pertain to the Bishops either to elect his successor, or to designate the electors: nevertheless, after the Pontificate of the world was joined to the bishopric of the City [posteaquam unitus est Pontificatus orbis Episcopatui Urbis], the immediate authority of electing in that case would have to be permitted by the bishops of the whole world to the neighboring bishops, and to the clerics of the Roman Church, which is proved in two ways.
First, because the right of election was transferred from all the neighboring bishops and the Roman clergy to the Cardinals, who are a certain part of the bishops and clergy of the Roman Church; therefore, when the Cardinals are lacking, the right of election ought to return to all the bishops and clergy of the Roman Church.
Second, because this is a most ancient custom (as we showed above from Cyprian), that the neighboring bishops, in the presence of the clergy, should elect both the Bishop of Rome and others also. And it is unheard of that the Bishops or Archbishops of the whole world should meet for the election of the Supreme Pontiff, except in a case where it is doubtful who should be the legitimate electors. For this doubt ought to be resolved by a general Council, as was done in the Council of Constance.

IMO the stronger argument can be made for the clergy of Rome (including the vicar general and auxiliary bishops) electing a new Bishop of Rome (who is Pope by definition), but then you have the issue of whether the clergy of other dioceses elect their bishops, or if those bishops are chosen by the Pope instead.  Clearly it is the latter.

But if you have no cardinals (as in the scenario described in the OP), who are basically the proxies or stand-ins for "the clergy of Rome", then there's really no one else to elect the Pope except (a) the clergy of Rome who are not cardinals, or (b) the bishops in college, a kind of extra-canonical conclave.

I don’t really want to argue for Orthodoxy on this forum because it looks bad to hijack a Catholic forum like that and I joined before I converted, but I do find the incessant assumptions of the worst from a lot of people on this forum to be dumb.  Not surprising, mind you.  I guess among my other reasons I listed which are more foundational I should say that I have no interest in being a part of a community that is this chronically suspicious, judgmental, and always assumes the worst of other people

sure “judge not lest you be judged” gets chronically misused by liberals, but I think some of this forum should take it to heart

Then there is 'By their fruits you will know them.' 

A sedevacantist website called the WM Review did a study of about half a dozen theologians who taught what should be done if there were no college of cardinals, or if they were somehow unable to elect a pope, such as by being locked in dungeons or something.

Hmm.  I was just asking elsewhere about something written by James Larrabee regarding papal elections without Cardinals, but I'm not sure if this was it.  For some reason, I don't remember it being a translation of something St Robert Bellarmine wrote.  I seem to recall it was his own thoughts.