People tend to shrink with age, especially after age 50 or so. It has to do with spaces between the joints amount of cartilage and posture. Bone length does not change.

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.

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.

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.

A rich diet does lead to earlier puberty but to my knowledge it does not student the adolescent growth spurt. Early pregnancy, before reaching adult height, does slow or stop growth.

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.

The second and third generation effect of immigration accounts for only a small portion of the height shortfall between Americans and northern Europeans. Besides the slowdown in American heights began in the 1950s, well before large-scale immigration into the country.