Adoption and twin studies mostly. I believe the wikipedia article on heritability of intelligence is still decent enough to be a good starting point.
This one? It does not indicate nearly the certainty that other commenters here do. Read it yourself, it's a good argument for what I'm saying. The GP characterized this scientific debate as settled, yet there is actually very slim evidence in that article for what is being claimed.
Why would it be surprising if among 1000s of the highest IQ individuals in some particular place, that there would be genetic similarities between them? It's completely possible that large groups among these people share a closer familial lineage far back into history that has influenced their present-day social and economic status, which are the factors that are actually responsible for developmental health and correspondingly intelligence, and so the genetic similarity is just a byproduct of the conferral of wealth and status between generations over a long period of time.
There are other problems with twin studies, most importantly that developmental health starts at conception which means that genetics are not the only thing that twins are sharing. Once again I'll refer you to Wikipedia for further criticisms of twin studies:
There have been many many papers from many angles on this over the years. The high heritability seems to be a well established fact.
Your third paragraph makes me suspect you don't know how twin studies often work. The idea is that you measure the IQs of genetically identical twins and compare how similar they are to how similar non-identical twins are. This lets you isolate what is purely personally genetic.
No it doesn't. Not at all actually.
https://en.wikipedia.org/wiki/Twin_study#Criticism
Maybe an actual geneticist can jump in here, but it's my non-expert opinion that you all are jumping to conclusions about a topic that you're not experts in either.
The very first critique is basically a nitpick. Yes, figures will be overestimated, but not taking into account air resistance when dropping your bowling ball off the tower of Piza isn't going to invalidate a conclusion that gravity is an accelerational force of at least 9m/s^2.
In fact, with the exception of the statistical critique by the guy with a giant axe to grind it seems mostly nitpicking. Dr Axe Grinder seems to mostly be critiquing the additive model, claiming it gives wrong results, but more recent papers (such as Polderman et al (2015)) seem to say that the additive model is mostly correct. And given the latter paper is a many-author meta-analysis rather than an emotionally charged rant by a single professor and is much more recent, I'm inclined to go with the latter paper.