Personally, I prefer to distinguish between "proof" (wherein a claim is necessarily true) and "preponderance of the evidence" (wherein the truth of the claim is evaluated in terms of probability). So my opinion on the subject is:
1a. There is no proof that Jesus existed.
1b. On the available evidence, the claim that there was a historical Jesus is not implausible, and is probably more plausible than the alternative.
Historical judgment rarely has recourse to "proof" on account of the nature of the evidence, which is always circumstantial and usually incomplete; the assessment of historical claims depends on corroboration of evidence, sociological and historical plausibility, parsimony of explanation, and the relative absence of reasonable doubt. Thus far I find the existence of Jesus (as an early first-century Galilean preacher with a following of disciples) no less plausible than that of Paul, or John the Baptist, or the Essene "teacher of righteousness" who founded the sectarian community at Qumran. Without going into detail, my opinion is based not as much on evidence pertaining specifically to a person named Jesus, but rather what is known about the social dynamics of early Christianity as a mass movement within Judaism and how it compares with other religious movements in Judea in the early first century AD which had distinct founders (e.g. Judas of Galilee in AD 6 founding the Zealot movement, John the Baptist in the 30s founding his own baptismal movement, a particular Samaritan prophet in AD 37 who started a Samaritan messianic movement, Theudas in AD 44-46 who started his own messianic movement, etc.). The data pertaining to James the Just in particular, and the kind of movement that he led at mid-century (see some of the discussion here: http://www.geocities.com/paulntobin/paulorigin.html), persuades me that he and his brother were real figures in the history of Second Temple Judaism.
Of course, proposing that there was probably (but not certainly) a historical figure with the name of Jesus at the root of the movement is quite different from attempting to define the life and beliefs of this individual, as many have done in scholarship without success. I am reminded of this comment by a historian about Alexander the Great:
"There are three Alexanders: the legendary Alexander, the historical Alexander, and the real Alexander. The first was born in men's minds soon after the death of the last, and he still lives in the East as Iskander. He has been many things, a saint and a devil, a defender of civilization and a barbarian, a perfect knight and a worthless debauchee. The historical Alexander is dead, but he is frequently revived in the pages of histories and biographies that fashion him in the image of each particular age admires; in one age he may be 'greater than Napoleon,' and in another he may be the man who first dreamed of 'one world'. The real Alexander died in Babylon about the thirteenth of June, 323 B.C.E. We know a little of what he did, but we shall never know what he thought or what he was like. The real Alexander is gone forever" (Tom B. Jones, Ancient Civilization, 1966, p. 283).
In the case of Jesus, we probably do possess more information of his thought and beliefs in the gospels, but attempts to delineate exactly what is genuine Jesus tradition and what is not fail for pretty much the same reasons given here. We are on much surer ground to start with the movement as it was when it was first documented in early Christian literature, rather than define exactly what goes back to the earliest stage and what does not (although I do have some thoughts on this).
I disagree with the view expressed in the OP on the value of the Bible as evidence. I understand and appreciate the point about blind allegiance to the Bible's presumed authority ("I believe the Bible because it says so"), but historical inquiry does not reject pertinent sources merely because they claim (or are claimed subsequently) to have some sort of authority. The various writings of the NT are still the oldest sources we have and they give invaluable information about the Christian movement in its early stages. It is certainly possible to utilize them critically without involving a sort of blind acceptance of whatever they say.
As far as claimed non-Christian sources are concerned, they are imho of limited help. You can find the texts here:
Josephus: http://www.textexcavation.com/josephustestimonium.html
Suetonius: http://www.textexcavation.com/suetoniustestimonium.html
Tacitus: http://www.textexcavation.com/tacitustestimonium.html
Pliny: http://www.textexcavation.com/plinytestimonium.html
Phlegon: http://www.textexcavation.com/phlegontestimonium.html
Thallus: http://www.textexcavation.com/thallustestimonium.html
Lucian: http://www.textexcavation.com/luciantestimonium.html
The brief mention of "Jesus the so-called Christ" as the brother of James the Just in Josephus is probably the one reference in all of this that is actually useful. The long testimonium elsewhere in Josephus is clearly interpolated, although it is unclear whether the whole thing is an addition or only a portion. After looking at the matter in detail, I am still not decided on what is more probable. But in light of Origen's positive statement that Josephus disbelieved that Jesus was the Christ, and especially the very odd context that the testimonium finds itself in (i.e. immediately before a digression on religious scandals from AD 19 that have nothing whatsoever to do with the career of Pilate), it is quite possible that the original testimonium was one in which Josephus negatively portrayed Jesus as a scandalous figure. The statement by Suetonius is vague and if "Chrestus" refers to Christ, it simply attests the fact that there were followers of Christ, i.e. Christians, in existence -- nothing about the historical Jesus per se. But the passage could also be interpreted as referring to someone else named Chrestus during the reign of Claudius who goaded the Jews in Rome into rioting. The statement by Tacitus is much better, as it specifically refers to the historical founder of the sect as someone who executed under Pilate during the reign of Tiberius. But his wording shows that he was not drawing on Roman sources, which would not have referred to Jesus as "Christ", or referred to Pilate as procurator. Tacitus attests what Christians had been loudly proclaiming for many years (cf. the similar statements in Ignatius, the Pastorals, and of course the gospels). Pliny also simply attests what Christians he knew said and did, and much the same thing can be said in the more satirical portrait in Lucian. And there are good reasons for thinking that the statements of Thallus and Phlegon (which are quoted secondhand) have nothing whatsoever to do with Jesus (see http://www.infidels.org/library/modern/richard_carrier/thallus.html).
