"Funny Aneurysm" Moment: In the last book, Mary lists Bill Cosby as one of the men that isn't evil while thinking about male violence. This was before the accusations against him were well known. It's especially ironic since she's a victim of rape, and this a large part of her character's arc throughout the books.
Mary Suetopia: The Neanderthal world has very little crime, hasn't had a war in hundreds of years, and lives in harmony with nature.
Deconstructed in later books. The Neanderthal society deliberately chose to breed violence out of its population because due to the size and strength of an average individual, a minor dispute could easily end with someone getting killed. One such scene is depicted in the first book; a dispute that two characters escalates into one having his mandible essentially destroyed with one punch-it's later stated that most past assassinations consisted of someone simply walking up to the target and caving their head in by their fists alone. The problem arose when they started sterilizing anyone sharing half or more of the violent person's genome as well. It's implied that the sterilization policy has actually created a pretty significant problem with domestic violence, since the victim can't report the crime without condemning any children they might have to the same fate as the abuser.
Sawyer has argued both in other novels and Real Life for something like the surveillance system of the series, feeling privacy is not only overrated but dangerous, and increasingly disappearing anyway.
The research available at the time the books were written indicated that Neanderthals and Homo sapiens were genetically incompatible, which plays a pivotal role in the plot. New research shows the opposite: not only were the two species compatible, there was significant interbreeding.
Dogs in the Neanderthal universe are physically indistinguishable from wolves. The reason given is that Neanderthals only selected for behavior and hunting companions, and did not care what their dogs looked like. Research with tame silver foxes has shown how selecting for tameness inevitably results in other neotenous traits (short legs, larger skulls, folded ears, curled tails), even without specifically breeding for them. With this in mind, Neanderthals would have had to actually put a great deal of work into breeding dogs that still resembled wolves in any way shape or form, and the primitive appearance of dogs would actually have had to be very deliberate.
The book's anthropologist character says there's no clear evidence for Neanderthals believing in an afterlife. Subsequently, this seems to be why Sawyer depicts them as uniformly having no concept of that, or gods. Since then, anthropologists have concluded that they did deliberately bury their dead with tools intended for use in another world.