It is possible to be rational about some things and irrational about others, just as it's possible to be right about some things and wrong about others. We all do it; we think of a shirt as lucky, we play the lottery despite the expected outcome, we approach the beatiful woman who we know will knock us back. No human being is rational all of the time.
Religion often invites and allows irrationality by discouraging people from seeking evidence or questioning doctrine. Scientists and other such intellectually driven people might even relish the chance to relax their faculties now and again, and be swept up in the community of a shared belief.
It is also possible to be both rational and religious because being rational doesn't mean being right. Incorrect premises in people's reasoning can lead them, perfectly rationally, to an incorrect conclusion. The premises on which religions base their truth claims are incredibly hard to pin down.
Your examples of rational theists are not good examples. Franklin was a deist for much of his life, and even as an elderly Christian held "some Doubts as to [Jesus's] divinity". Einstein did not believe in a personal god, or anything which could contravene the laws of the universe. He effectively thought God WAS the laws of the universe, and this is to what he referred when he used the name. He was something like a pantheist, and basically said so: "I believe in Spinoza's God, who reveals himself in the harmony of all being." I encourage you to read up on Franklin, Einstein and Spinoza. They're fascinating people.
There are a great many issues with the truth claims of Islam. I suggest starting by searching Wikipedia for "Historicity of Muhammad" and working from there. Do not take Wikipedia at face value, of course.
Unfortunately fundamentalist Muslims, like any other religious fundamentalists, will not generally accept criticism of their religion. Muslims who lose the faith, apostates, are marked for death. Non-Muslims are infidel and not to be trusted. You can see how it's difficult to attain a platform from which you'd actually be listened to.
Don't take it for granted that if Muslims were presented with evidence that their beliefs are false, they would abandon them. It would be far easier to believe that the evidence itself is false or, even more cunningly, a test of faith.
Nobody wants to be a traitor, but that's currently the only official way to leave Islam. Fundamentalist Islam especially has insulated itself against Muslim and non-Muslim reason alike. It'll be tough to inject any.
First, I think it's important to understand the incredible amount of information one has to look over tediously before they can reach a conclusion of a historical or ahistorical nature. The truth is, I've been researching this subject for seven years and there are things I'm just starting to skim over now.
Now that that is out of the way, let's tackle your questions. You ask, "What evidence is there to support that Jesus never existed?" First and foremost, if you can provide me evidence to support that fairies don't exist, I'm all ears. One can't ask to disprove a negative, because there is nothing to disprove. One must look at the available evidence that already exists (or doesn't...as in this case) and determine if that evidence is sufficient to establish historicity.
To help you better understand this lets use an example given by Christians where they assume that we mythicists assume the historicity of somebody famous without evidence. Aristotle is usually totted around the most by some ignorant or misinformed person as having no contemporary evidence of his existence - as a standard if you will to suggest that Jesus should be considered to be on the same level of accepted historicity as Aristotle. However when comparing the list of evidences between the two, there is no compatibility. Here's a brief list of the differences between Aristotle and Jesus:
And this is not the half of it. Aristotle not only wrote tomes of prose in his time, by his own hand, but also contemporary accounts exist of Aristotle. As Richard Carrier states on Aristotles contemporary accounts, "There is one fragmentary inscription dedicated to Aristotle still extant at Delphi that I believe was erected in his lifetime. We have substantial portions of the Elements of Harmonics by Aristoxenus, a contemporary of Aristotle, which mentions him briefly. Anaximenes of Lampsacus (not the presocratic of the same name), also a contemporary, wrote an Art of Rhetoric that survives, and it addresses Aristotle. Theophrastus was his pupil and contemporary and we have some few of his writings, but I don't know off hand if they mention Aristotle by name. Isocrates was his contemporary and sometimes opponent and he may have mentioned him, too, but again I can't say for sure if he ever actually names him in extant works. There was certainly a great deal of contemporary writing about Aristotle, but as far as I know little to none was preserved, except in later sources. TLG shows a few hundred contemporary, named references to Aristotle, which are cited or quoted by later authors." Carrier also suggested a book, "Lloyd's book "Aristotle" would probably say what else there is."
This is vital because we have NO accounts of contemporary evidence for Jesus. None. The earliest extant manuscripts for Jesus date to Paul, thirty years after Jesus supposedly died, written by a man who never met Jesus, knows nothing about him, or about any of his deeds, or miracles or speeches. Paul doesn't attribute any words to Jesus nor does he seem to - in any fashion - refer to Jesus in a physical, literal sense.
After Paul, we have a forty year gap of nothingness. At the very end of the first century CE, we have rumors (just rumors) of hearsay about a being Jesus. The earliest Gospel fragment we possess is the P52 fragment, and it's barely a scrap of parchment from what appears to be John. But it's too weak a source to use to compare. That is it. And when is this P52 fragment from? 130 CE and no earlier. That's a hundred years after the supposed death of Christ. Now here's the funny part, we have works from Aristotle that survived from 500 years before THAT, and yet we can't find one contemporary account of a man who is said to have walked on water, and preformed all these miracles, or even rose from the dead?
I hope that helps you understand a little bit as to the problems associated with the question of whether there can be a way to disprove a negative, and also with understanding what sorts of evidence is looked for when trying to determine historicity. Your second question is far easier to answer, as I have already written extensively on the subject.
At this link HERE you will find all you need to know about the supposed evidences for Jesus and why they hold no water. You can ALSO check out THIS LINK for additional information on the Mythicist position and the case against a historical Christ.
Oh, by the way....welcome to the campaign.
Rational Response Squad Co-Founder