2 Timothy 3:16,17, & 2 Peter 1:21 explains Gods involvement in the holy scriptures. But with regards evidence of Gods involvement in Genesis and the entire bible, well the evidence is available on a number of levels. Historicity, scientifically, archaeologically, and prophetically I have come to the conclusion from the evidence God most certainly did inspire men to record what he wanted recorded for our prosperity. If after considering the evidence you come to a different conclusion, well that's your business.
First off, if the sources for inspiration are evidences for god, then that would mean that there would be evidences for every single gods and goddesses in the ancient world, from Greece, Egypt, Sumer, India, Japan, etc, because the deities of these religions have been sources of their inspirations.
If inspirations, then anyone writing about ghosts, fairies, elves, dragons, phoenixes, and whatever mythological and fairytale creatures or beings would evidences too.
And if inspirations were equaled to evidences, then I might as well as believe in Asgard and (from Norse myths), or Mordor of Middle-Earth (from The Lord of the Rings) or Hogwarts (from Harry Potter) for places that existed, to be true.
Do you see where I am going with this?
To use inspirations as evidences would be bring whole lots of things we read about as evidences.
Second, if you are talking about the bible naming some real cities and kingdoms within the bible, doesn't mean much as evidences for the existence of your god, nor what they write myths about certain people, like Abraham, Moses, David, Solomon, just to name a few, doesn't make the biblical stories to be true.
In Greek, Egyptian and Sumer-Babylonian myths, they named real cities all the time in their traditional stories, but that doesn't make their gods or their myths to be true.
Two of mine all-time favourite books are - The Iliad and The Odyssey. Homer named many cities and kingdoms in Greece and in Asia Minor that were real, including that of Troy, Argos, Pylos, Knossos, Ithaca, etc, all of which did exist in the Bronze Age, as well as in his time. He also named kings to these kingdoms and cities, but does that make the stories real?
Many people today write fiction, giving details of cities in which the stories are set in. The cities may be real, but stories and characters are more likely fictional.
You have no way to link your God to cities or kingdoms, because the god-part, is merely YOU projecting your belief and faith in a book full of myths to actual location, doesn't make it "archaeological" evidences for the existence of your god, PERIOD!
Evidences are something that you can verify to be TRUE or FALSE, and there are no evidences for actual god to exist. What you do have evidences is that there are religion(s) that you have built around this particular god, but no evidences for the god, and none for heaven and hell, angels and demons. All the deities, angels and demons, are just superstitious BELIEF - nothing more, nothing less.
Lastly, there are no (scientific or archaeological) evidences for Genesis' 6-day creation, or man being made out of dust or soil of the earth, the flood, or Tower of Babel (in which mankind spoke only one language one day, and many languages the next), the life and adventures of Abraham, Isaac and Jacob, the mass-exodus of Israelite out of Egypt and their invasion of Canaan (in book of Joshua), etc. All of these are myths.
As to Jesus, there may have been a teacher who roam about Galilee and Judaea, as being a real person, but all those miracles (including the last one - the resurrection) the gospels claimed to have witnessed, seemed highly unlikely, and may have turn possibly a real person to that of legend status. Jesus may have taught his disciples and to the public in parables, but the miracles seemed to be just make-believe hoaxes, created by people who may have never met the real Jesus.
This "people" I am referring to, are the authors of accepted gospels in the New Testament. The gospels were written anonymously, but in the 2nd century Christians have attributed names and traditions to the gospels. We don't know who the real authors to those gospels are, but it is highly unlikely they met the real historical Jesus.
I think your problem is that just because it was written in the bible, you think the scriptures are telling you the truth.