I think the answer is complicated. Homo erectus, the first homo species thought to use fire and our direct ancestors were as close to obligate carnivores as there is in the homo genus, but they focused on big animals with a lot of fat like hippos and elephants. They likely did not cook that fat, because it would store just fine without doing so.
I could only recommend an introductory college course since that's where I picked up the basics. Hopefully someone else has better resources. An important thing about sociology is understanding the different approaches there are to things and the language those approaches use.
Don't worry about it. If you were really wrong someone would chomp at the bit to reply to you about how wrong you are. If they're not, you either have an unpopular or popular enough to be spam, opinion.
Oh for sure. We already had complex social relationships that involved lying when we were only homo erectus and likely incapable of speech and were hunting full grown elephants and hippopotami(yeah, simple stone tools against those monsters required some serious teamwork). I think that creating a social face for those you DON'T know, though, had to come about once we were in a situation where there were people we interacted with that we didn't know. Hunter/gatherer societies generally still operated with too much of a cohesion for you to truly be "fake".