basic emotions such as, disgust, surprise, fear, etc.
Genetic. Rather simple animals clearly exhibit these as well. Everyone has seen an ant, a roach or a housefly fleeing in panic.
avoidance of hazards (heights, disease, etc.)
Even insects avoid hazards.
face perception
I voted for this, but I'm actually not sure anymore...
This could be must-be-learned. But, well, babies recognise
and
very early. Maybe it depends on the expression and how subtle it is.
notion of good and evil
About the same things are considered moral or immoral in different cultures (lying, killing, stealing, adultery...) even when they haven't been in any contact with each other. There are, of course, some exceptions, like cannibalism that is actually normally avoided by most animals.
notion of fairness
Animals exhibit this. I have seen countless and countless times a horse demanding justice by neighing, snorting and stomping in pure ANGER as his neighbor gets his daily hay but he doesn't
notion of altruism (concern for the needs of others)
I don't think so. I... I just don't think so. (I blame my iNtuition for this!)
notion of God(s)
I think so, yes. All cultures do have and have always had gods and religion. Today's religions are money, science, careers and so on.
mathematical truisms (obvious mathematical truths)
I've read that horses, dogs and cats can naturally count somehow like "one, two, three, four, five, a lot".