Luke was written in the 70's, 80's or even later, how is this relatively shortly after 'Jesus' death and 'Ressurrection'?
Because, without printing presses, it took years for works to be produced. Plus, your dates aren't quite up to snuff, as traditional Biblical scholars have Luke written earlier around the 50s A.D.
Christians love saying this, especially in light of guys like Julius Caesar who DO have loads of contemporaneous testimony. 1st century CE is a big pond. Considering that Mark, ca. 70 CE is the earliest, this hardly qualifies for 'contemporaneous'.
Traditional scholars have Mark's Gospels in the 50s A.D. But even with that late of a date, writing about a historical figure, even with that late of a date (70 A.D.) is hardly problematic and still does nothing to deter the existence of Jesus Christ as an actual person. Having loads of "contemporaneous history" vs. having not as much does NOT equate to the latter someone not existing.
Yes and we have extremely detailed information of all those years just prior to Jesus' magic ministry. 
Exactly how is not detailing the life of a carpenter, during His teenage years, a sign of someone not existing? You're grasping at straws, again.
Many scholars have argued that, even if the Annals themselves were genuine, the passage regarding Jesus was spurious. One of these authorities was Rev. Taylor, who suspected the passage to be a forgery because it too is not quoted by any of the Christian fathers, including Tertullian, who read and quoted Tacitus extensively. Nor did Clement of Alexandria notice this passage in any of Tacitus's works, even though one of this Church father's main missions was to scour the works of Pagan writers in order to find validity for Christianity. As noted, the Church historian Eusebius, who likely forged the Testimonium Flavianum, does not relate this Tacitus passage in his abundant writings. Indeed, no mention is made of this passage in any known text prior to the 15th century.
The tone and style of the passage are unlike the writing of Tacitus, and the text "bears a character of exaggeration, and trenches on the laws of rational probability, which the writings of Tacitus are rarely found to do." Taylor further remarks upon the absence in any of Tacitus's other writings of "the least allusion to Christ or Christians." In his well-known Histories, for example, Tacitus never refers to Christ, Christianity or Christians. Furthermore, even the Annals themselves have come under suspicion, as they themselves had never been mentioned by any ancient author.
It is a peculiar and disturbing fact that the entire Annals attributed to Tacitus never existed until their discovery by Johannes de Spire, at Venice in 1468, and that this sole copy, purportedly made in the 8th century, was in his possession alone. The history of the Annals begins with the Italian calligrapher, Latin scholar and Papal secretary Gian Francesco Poggio Bracciolini (1380-1459), who, writing in 1425, intimated the existence of unknown works by Tacitus supposedly at a Benedictine monastery in Hersfeld, Germany. "The Annals" was subsequently "discovered" in a copy of Tacitus's Histories at the monastery, in the sixteenth century. This text was not named "Annals," however, until 1544, by Beatus Rhenanus.
In 1878, the "excellent Latin scholar" WJ Ross wrote the book Tacitus and Bracciolini, which evinced that the entire Annals were a forgery in very flawed Latin by Bracciolini in the 15th century. Ross's work was assailed by various clergymen, who claimed the main defect in his argument was that "one of the MSS. [manuscripts] of the Annals is at least as early as the XI century." In reality, the critics had not actually read Ross's book, in which Ross does indeed address this purported 11th century manuscript, which he shows was merely pronounced by dictum to be early. Interested readers are referred to Cutner and Ross's books for further discussion of this debate, which includes, in Ross's dissertation, a minute examination of the Latin of the Annals. Suffice it to say that the evidence is on the side of those who maintain the 15th century date, in that the Annals appear nowhere until that time.
In any event, even if the Annals were genuine, the pertinent passage itself could easily be an interpolation, based on the abundant precedents and on the fact that the only manuscript was in the possession of one person, de Spire. In reality, "none of the works of Tacitus have come down to us without interpolations." Drews considers the Tacitus passage in its entirety to be one of these forgeries that just suddenly showed up centuries later, and he expresses astonishment that "no one took any notice during the whole of the Middle Ages" of such an important passage. Says he:
No one, in fact, seems to have had the least suspicion of its existence until it was found in the sole copy at that time of Tacitus, the Codex Mediceus II, printed by Johann and his brother Wendelin von Speyer about 1470 at Venice, of which all the other manuscripts are copies.
The reason for this hoax may be the same as the countless others perpetrated over the millennia: The period when the Annals were discovered was one of manuscript-hunting, with huge amounts of money being offered for unearthing such texts, specifically those that bolstered the claims of Christianity. There is no question that poor, desperate and enterprising monks set about to fabricate manuscripts of this type. Bracciolini, a Papal secretary, was in the position to collect the "500 gold sequins" for his composition, which, it has been claimed was reworked by a monk at Hersfeld/Hirschfelde, "in imitation of a very old copy of the History of Tacitus."
Regarding Christian desperation for evidence of the existence of Christ, Dupuis comments that true believers are "reduced to look, nearly a hundred years after, for a passage in Tacitus" that does not even provide information other than "the etymology of the word Christian," or they are compelled "to interpolate, by pious fraud, a passage in Josephus." Neither passage, Dupuis concludes, is sufficient to establish the existence of such a remarkable legislator and philosopher, much less a "notorious impostor."
It is evident that Tacitus's remark is nothing more than what is said in the Apostle's Creed--to have the authenticity of the mighty Christian religion rest upon this Pagan author's scanty and likely forged comment is preposterous. Even if the passage in Tacitus were genuine, it would be too late and is not from an eyewitness, such that it is valueless in establishing an "historical" Jesus, representing merely a recital of decades-old Christian tradition.
And, now you're back to your conspiracy theories again, all with the same tired theme, "The passage is legit, unless it talks about Jesus". This is pitiful.
Were you just complaining, about Josephus' work, that there weren't enough Christians to fit Josephus' description of them as a "tribe"? Yet, they somehow have enough tradition for it to be recited to the point where Tacitus would use it, in reference to describing Christians.
And, whoever you got this stuff from, he's dead wrong about one thing: Christianity hardly rests on Tacitus' account. The irony of it all is that Tacitus is actually commenting about on how Christians have spread.
I'll get to Josephus in a separate thread.
I've already seen it; and, it's nothing that helps your cliam one bit. The bottom line is that the historical reference that shows Jesus' existence, outside the Bible, are valid.
It's the same trend. First, you claim there is no claim of Jesus' existence, outside the Bible; when one is shown, you state it's a fraud; when the next appears; that's a fraud (or it's legit, except for the part that talks about Jesus, of course

).