>Your height can be significantly stunted by poor nutrition during youth, doesn't that mean that height is not genetically determined
No. Because it is hereditary to a statistically very significant degree. See pic related, if intelligence was completely nurture and not hereditary that would mean that people of lower intelligence would have grown up in absurdly horrendous environments of nutritional deficiency / starvation which obviously isn't the case in wealthy countries where we still see the differences.

You can measure human intelligence with almost any test / achievement / work, however all of these are also affected by effort, study, and hard work. Intelligence tests are designed such that the effects of everything but intelligence are minimized but they cannot be completely eliminated.

I think that what a lot of people get wrong about intelligence is that they think that it is somehow difficult to measure, when in fact it is extremely easy to measure. Just look at any personal achievements that require intelligence and compare to other people and you can make a statistically very good guess at a persons intelligence.