Is the probability of having children with a hereditary trait (red mottled) considered a legitimate Shar'i justification for annulment of the engagement, knowing that this trait exists in the girl's paternal uncle? And is there a sin incurred by rejecting the girl because of that, or by marrying her and having children with that trait?

If the mentioned trait is a defect and there is a fear of its transmission to offspring, then there is no legal impediment to considering this when choosing a wife. The specialized doctors should be consulted to determine the likelihood of its transmission. There is no blame upon you for leaving this girl and seeking another, and no sin upon you in doing so. Similarly, there is no impediment to her marriage even if the possibility of her offspring inheriting this trait is proven. As for the hadith (definition), "Choose carefully for your progeny, for indeed the root is tenacious (or: genes are insidious)," it is weak with this wording.