Another Christian who believes in evolution here! We had a straw poll in MN religion section a year or two back - I think it came out about 3/4 of the Christians believed in evolution, and 1/4 didn't. Though there were some grey areas in the middle with ID or very God-led evolution....
As to Neanderthals, I believe there is now a possibility that they will be classed as a subspecies of Homo Sapiens, as there's a question of whether there was some mixing of the two when Homo Sapiens expanded out into areas where there had only previously been Neanderthals. And, if the two "species" did interbreed, then they can't be different species, pretty much by definition!
To my mind, if they were self-aware then they could have a meaningful relationship with God just the same as Homo Sapiens could.