It is worth pointing out that archeologists no longer rely on the Bible except possibly in passing. It's been proven wrong too many times.
You'll forgive me if I haven't the time to look up a list just yet.
You can show some things are impossible, though.
As you mentioned, Jesus could not possibly have been born at the Time of Herod the great
and of Cyrenius as governor, since Cyrenius was appointed governor
after Herod had died.
DOC can choose one, or the other, but then he's stuck with a gospel that gets a basic fact of Jesus' birth wrong.
But for both to be right would be impossible, and hence at least one must be false.
And then there's the incident with the pigs.
Look at a map of where this supposedly took place. It's a village called Gadarenes (Mark 5:1), also referred to as Gerasa.
Click for link to a map
It's there, just North of the Jabbock river.
Tell me if you see any cliffs nearby. Or any sea.
Plainly, this story is either false or so heavily edited as to be indistinguishable from false.