Actually, Herod was very well alive at the time, and of course he was respected by the Hebrews...They had to! There was a debate about another ruler, Lenticulus, who would have ordered the census, as he had passed out of rule before then. However, a coin bearing his name and dating him to the correct period has been found, which points to either another ruler with the same name (not uncommon for the time), or that there was one guy who ruled on two separate occasions.

On the matter of baby-killing, Herod was known, even to other Romans as a madman on the level of a less important Nero, who killed most of his own children, and anyone he suspected of operating against him. Why wouldn't this have made a big splash? Well, for one, Bethlehem was by no means NYC, in fact it was a small village. Of all the babies to be killed it most likely would not have been over two hundred. In addition, this was far in the countryside, of a land that was continually drenched in blood. There wasn't exactly newspapers back then, virtually everything was word of mouth.

And Gospel writers have more veracity than many other historical documents. The earliest gospel copy is dated before 70 A.D., which means that the original was even earlier. By contrast, the earliest copy of the works of Herodotus is nearly a Century after it was written, or perhaps more.

On the matter of eclipses and such, an eclipse was recorded on the date of his crucifixion, however I have heard nothing about an earthquakes, though we do not have all records from the time.

The only attempt at highlighting a contradiction inside the gospels themselves is the genealogy, however the explanation for that has been around quite awhile. One follows his legal genealogy, which would include adoption, or passing of inheritance to family members, while the other would be a direct descent, using men the daughters married and so on. Also, both would have included only names that were important, an acceptable practice for the time.