[BroJoeK]

varmintman: "The Neanderthal has been abandoned as a plausible evolutionary antecedent for modern man precisely because the genetic gap is too large (DNA halfway between ours and that of a chimpanzee)."

Your claim has been posted and corrected here before.
Here it is again, in a nutshell:

• Any two humans on average are different by about 3 million "base pair" mutations -- that is .1% of 3 billion total base pairs.
  Of course, biologically all humans can interbreed with each other.

• Humans and Neaderthals' DNA differs by about 9 million "base pair" mutations, which is .3% of 3 billion total base pairs.
  DNA analysis shows about 4% interbreeding between Neanderthals & Europeans, down to 1% with Asians, but zero with Africans.

• Depending on exactly what and how you count, human and chimp DNA differs by 50 million to well over 100 million "base pair" mutations.
  Of course, humans and chimps can not interbreed, under any circumstances.

Bottom line: Neanderthal DNA was over 90% closer to humans than to chimpanzees, indeed Neanderthals were close enough to interbreed with Europeans, which means they were not a separate species at all, but rather a sub-species of homo-sapiens.

varmintman: "All of the creatures you mention other than for the Cro Magnons were hominids, i.e. bipedal apes..."

The word "hominid" refers to all great apes -- orangutans, gorillas, chimps and humans -- who branched from common ancestors around 14 million years ago.
The word "hominid" also refers in a more restricted sense to humans and pre-human relatives more closely related than chimpanzees.

varmintman: "Anything 300K - 500K years back which anybody could try to claim was a "common ancestor(TM)" to both us and the Neanderthal (Usually given as homo Heidelbergensis)", would be much more remote from us THAN the Neanderthal.
Too-genetically-remote-to-be-ancestral-to is a transitive relationship "

All modern Europeans and Asians are recognized today as having migrated "out of Africa" beginning around 125,000 years ago.
So no pre-existing pre-human populations in Europe or Asia were our ancestors, but there is DNA and fossil evidence of some interbreeding with Neanderthals.

varmintman: "Another is this new claim of 1 - 4% Neanderthal genes in everybody other than Africans.
Once again, the Neanderthal was a glorified ape.
Any crossbreeding with a glorified ape PRIOR to the bottleneck and Africans would not get left out.
Any crossing AFTER the bottleneck and not involving Africans as claimed, and the genetic gap between Africans and everybody else would be gigantic, rather than minuscule as it actually is. "

We are "glorified" apes -- hominids.
Neanderthals were rather more humble, I'd think.

Whatever interbreeding happened as modern humans migrated out of Africa into Asia and Europe, was relatively minor, leaving traces of 1% to 4% in our DNA.
This suggests an occasional "boys night out", but, but, which were the "boys" and which were the "girls" they went to visit? ;-)
More important, why did their parents allow these offspring to live and grow up?

[varmintman]

Images courtesy www.themandus.org, what Neanderthals actually looked like:

Without the ice age fur coat, for illustration purposes:

There is zero genetic evidence of any relationship between humans and Neanderthals.

and there is zero evidence of crossbreeding in the Levant despite the two groups living there in close proximity for long periods of time.

Neanderthals were smart, and they did not lack for courage. They were big game hunters taking on mammoths and other large ice-age prey animals with thrusting spears, but their only real form of interaction with humans was eating them, and they paid the price for it.