I have heard arguments for the existence of the Christian god based on aspects of human psychology which could seem to have a supernatural origin, such as reasoning, morality, and emotions. It is suggested that these traits must be god-given, cannot have evolved naturally, and will never be explained from a purely biological perspective.

Do any Christian theists here agree with this view? If so, how do you feel about disciplines such as affective neuroscience, which explores the biological basis of emotion? Specifically, if I told you that other primates can act morally or that dogs have emotions, would this challenge the idea that such traits must be god-given (I don't think the bible mentions god giving moral reasoning to monkeys)?

Yes.

Many may say "yes" to this now, but then again, many said "yes" to the germ theory of disease and suddenly that's no longer a problem for theology.

It's an issue which meets several "limits":

1: There is nothing to indicate cognitive faculties originate from anywhere but the brain and body. There is even a strong case to be made that people who no longer possess their brains aren't very emotional.

2: If we use reductionism we will in psychology, like in any other scientific discipline, ultimately reach a level we can't explain. If we go by physics we will in any discipline even ultimately reach a level we can't measure (below planck length, below planck time). This does not imply supernatural, it implies no knowledge.

1 implies that that there is no need, research-wise, to assume intervention beyond our own brains or bodies. 2 implies that emotion being ultimately inexplainable is a trait it shares with bricks.