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.

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.

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

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.

There are some effects of the biome. Compact bodies tend to conserve heat relative to linear ones. This can affect average height as well as body proportions. The study participants Europeans and Americans do not fall into the range where this effect is powerful, however.

Africans are not as tall as northern Europeans. But Manute Vol and other sports figures in basketball and football receive a lot of press coverage.