Both are minority views at present. The view that Christianity was a competing mystery religion (albeit a Jewish version) is more widely accepted (as that does not entail Jesus didn’t exist; it doesn’t even entail that he invented those aspects of the religion). For comparative perspective, the view that Jesus didn’t exist is more widely accepted than the view that Jesus was an armed militant (which is nevertheless a respected hypothesis taken seriously in the field).

It is difficult to know how widely this view is granted because no one polls these beliefs among experts, almost no experts have even read the peer reviewed studies questioning the consensus (and thus are in no way capable of evaluating them until they do read them), and several experts have expressed worry about “outing” themselves as admitting to either view (there is a lot of peer pressure in the field against it). Moreover, more than half the “experts” are actually Christian apologists who cannot admit these things (neither personally, as it would destroy their faith; or professionally, as it could get them fired, demoted, censured, ridiculed, or harassed). So we are not likely to get an honest or open accounting of even what the actual consensus us, or who is pushing it. I discuss this problem in my forthcoming book.

But for now, we do have a list of scholars who have gone on the public record admitting doubts that Jesus existed or admitting such doubts should be taken seriously. It numbers several dozen now and is growing: List of Historians Who Take Mythicism Seriously.

Invented pedigrees was a common thing back then (it happens in philosophical and medical schools of thought back then, too). And yes, all for the reason of trying to establish authority over competitors (hence competing Christian sects had their own pedigree lists competing with the one we know; in fact that one appears to have been invented to combat those).

There is no single book on this. You have to pick a specific sub-topic. For example, Moss on the Myth of Persecution. On the ancient development of the papacy there is Eamon Duffy’s Saints and Sinners: A History of the Popes, 4th ed, and Christopher Lascelles, Pontifex Maximus: A Short History of the Popes. There is also a handy survey of the evidence and its weaknesses by an amateur, Brandon Addison, that is reasonably restrained methodologically and useful for being thorough in citation.

On Paul and Peter: they are probably real, because we have two eyewitnesses attesting them (Paul for himself and Peter; and Clement for both). See The Historicity of Paul the Apostle and How Do We Know the Apostle Paul Wrote His Epistles in the 50s A.D.?; and for Clement, How We Can Know 1 Clement Was Actually Written in the 60s AD and Interpreting 1 Clement’s Supposed Descriptions of Fabulous Murders. The relevant letters bear numerous hallmarks of authenticity and contemporaneity. The evidence is sufficient to render their odds on existing at no less than 100 to 1. And that’s without Acts, which is wholly unreliable for this (see How We Know Acts Is a Fake History).

What about Paul and Peter – are they established as actually existing, or could they be mythical or dubious figures as well? What’s the probability they really existed?

No.

The word “pope” is a late invention. It does not exist as a thing for centuries (see Wikipedia). And the Epistle of 1 Clement makes clear no such position existed, either. Churches were autonomous, and one church could only hope to persuade another in the usual way; they could not cite rank at them. See my discussions in How We Can Know 1 Clement Was Actually Written in the 60s AD and Interpreting 1 Clement’s Supposed Descriptions of Fabulous Murders.

Moreover, the church was not unified until fourth century imperial decree (and even then, only on paper). So there was no single person to call upon. We do not know if Clement even was the bishop (rather than merely an elder) at Rome (in fact the name is not in the letter and it appears to be written collectively by a committee), but even if he was, there may have been competing churches at Rome who didn’t recognize that one; and there certainly were within a hundred more years.

So there was never any single hierarchy to identify. The one that prevailed did so only by state force later on. And that one had no such position as “Pope” until the third century. Their decision to rank the bishop at Rome as the top rank of the distributed church was late, and competed with, for example, the Orthodox Tradition, which eventually declared their “Patriarch” in Constantinople as the top rank (eventually splitting the Church again). And there were others who likewise split (like the Copts and Ethiopians).

Thanks! Who was the first pope who we know existed, or at least can be fairly sure? Clement?

Not really, no. Almost all that data (insofar as we even have any) is late and legendary, and unsourced and thus unreliable. What little we do have is extremely limited (e.g. in authentic letters Paul says a guy names Apollos was one of the Apostles, so that’s probably true, although he tells us next to nothing about him; likewise he identifies Cephas/Peter as the first Apostle and thus founder of the Christ movement, but then tells us barely little more than he does about Apollos; etc.).

But it’s not peer reviewed, it’s whackadoo, its author has no evident qualifications, and it is solely in defense of a bizarre sectarian religious view, not a reasonable historical conclusion. So it’s useless.

There are tons of things like that online. So why care about this one?