Taller people tend to earn more on average, although the effect is modest. This is clearly true in occupations where strength is important but also a relationship exists in other occupations as well. It could be perceptions, as you say or something systematic. It's somewhat of a mystery.

Young children from Guatemala who are adopted into the United States gain considerable height relative to the standards. So the height difference cannot be genetic.

The English were relatively short in the mid-nineteenth century and so their expeditions return with stories some exaggerated, not doubt about "giants" who lived in other parts of the world.

All the Americans in the study were native born, but the height of the mother does affect the growth potential of the child. So, there is a second generation effect. But this can explain only a small portion of the height difference between Americans and northern Europeans. Regarless of continent of origin, children who grow up under conditions of good health and nutrition are about the same height, on average.

Being shorter or taller depends on details of the diet. Vegetarians can obtain a balanced diet, with sufficient complete protein, vitamins, minerals and micro-nutrients. It's trickier for vegans but it can be done.

Genes are very important at the individual level but these differences tend to cancel at the national level.

I agree that zinc is one of the important micro-nutrients but people require a balanced blend of food for our biological engines.