Creationists researchers are also able to provide evidence of creation, a deceent back to 2 individuals, and no common ancestor with any ape. It does not matter that this evidence is refuted by evo researchers as they refute each other all the time yet still manage to agree 'it all evolved'.
This doesn't have anything to do with the MRCA between humans and other apes like chimps and gorillas, but I believe you mentioned this as part of an argument that scientists are all over the place regarding human ancestry and really know very little concrete. While I was able to find that there are disagreements about how fast the molecular clock "ticks", I wasn't able to find anything indicating that conclusive evidence for any particular figure, such as your claimed 6000 years, had been uncovered.
I think many scientists would agree that there's insufficient basis for claiming any particular date for mitochrondrial Eve or Y-chromosome Adam. Certainly the Wikipedia article on Human mitochondrial molecular clock supports that view.
It would be fascinating if 6000 years ago as the time of the MRCA were nailed down as the correct figure, but realize that MRCA stands for the Most Recent Common Ancestor and not for When the Species First Appeared. In other words, no matter how long ago the MRCAs, they weren't the first humans.
The more closely related are two species the more likely it is that they'll be able to interbreed. When two populations of the same species become isolated from one another (which is one way the Neanderthal species could have come about) then they will evolve independently and, of course, differently. The human ancestral population that migrated north from Africa evolved to become better adapted to more northern climates and to whatever else was unique in that environment, and the same was true of the human ancestral population in Africa. The more northern population evolved into the Neanderthals, the African population evolved into Homo sapiens.
The two populations never evolved far enough apart to become genetically isolated (unable to interbreed), so there would have been interbreeding wherever they came into contact with one another. This is true no matter when the split occurred, but the Wikipedia article on Neanderthals puts the split around 350,000-400,000 years ago.
Species doesn't have one unambiguous definition. By the definition "can't interbreed" Neanderthals and humans are the same species. By the definition "can interbreed but not usually", Neanderthals and humans are different species, in the same way that tigers and lions are different species, as well as zebras and horses.
The label isn't important. Humans and Neanderthals could and did interbreed, but not a lot. Whether you want to call them the same species or not is just terminology.