While bigfoot may be based on primates, known primate species don't live in areas like North America or other places with stories of similar creatures, so the stories aren't based on local primates, accept humans of course. Bigfoot legend based on hairy men? It's not just Bigfoot, goblins, sea serpents, are present in many cultures, although they tend to have different attributes according to different cultures. Given the differences, they're not as universal as you seem to suggest, and I a little skeptical about how universal you claim them to be.
A lot of religions have the golden rule and "don't kill people" in their teachings. A lot of non-religious organisations have a "don't kill people" policy as well. Arriving at the golden rule seems independent of gods.
Every culture has some higher power concept, or agency or agencies controlling things in the background, for good or evil. Humans seem biased toward agency as an explanation. One explanation could be that there is an agency but if you look at religion that's not really going to give you an idea of what that agency is, there's quite a bit of variety. I don't think you can get geniune universals from all religions without becoming meaningless like "higher being" which isn't a good description at all.
Another explanation could be that bias towards an agency evolved as paranoia that was an evolutionary advantage. If you're paranoid, sometimes you're going to be right. Conspiracy theorists were right about Iran-Contra, but that doesn't make them right about other theories. If you're an ape, being paranoid and wrong most of the time might be a disadvantage in some enviroments, but might be a big advantage in other enviroments. You can be right once out of a hundred, but that once might be really important to your ability to replicate your genes.
Humans are biased towards bad logic, especially when young, but even really intelligent experts in other fields get statistics terrible wrong if they've never been educated in it. So innate qualities in humans to believe in certain things, I think are most likely explained through psychology, not through theology.