I think your explanations are largely incorrect. I’m currently partway through reading The Bell Curve; it investigates a number of things people often attribute to socioeconomic status and tries to compare the effects of socioeconomic status and intelligence.
An IQ score is literally your score on an IQ test; I think it’s more correct to talk about g (“general factor”) which is the principal component of intelligence you get if you measure a group of people on a battery of tests, and is the most predictive way to summarize an individual’s mental ability in a single number. A good IQ test is highly “g-loaded”, that is, it does a good job predicting someone’s scores on other tests that are g-loaded to some degree.
Intelligence, as with most traits, is influenced to some degree by genetics and to some degree by environments. This can be (approximately) measured by comparing differences in intelligence between people of varying degrees of relatedness (somewhere between being identical twins and being randomly selected people), who grew up either in the same families or separated. Usually people are interested in pairs of siblings or identical twins where one was raised by families of different socioeconomic statuses. When The Bell Curve was written, the authors said the consensus was that intelligence is between 40%–80% genetic, but I think it’s now thought to be on the high end of that. Intelligence is actually more influenced by genetics in adults than in children, which is the opposite of what you’d expect if environment actually did have a large effect.
When you say you can “study” for an IQ test, I think this is kind of tautological; e.g., you could memorize the answers to the hard problems, but intelligence is largely stable through life. It’s true that IQ scores have been rising for several decades (the Flynn effect), but I don’t think it’s ever been explained really well (maybe fewer people growing up in damaging environments?), it’s not clear whether or not people are actually getting smarter or they were just scoring better for some reason, and I think there’s evidence that the trend is starting to stall or maybe even decline since the mid 1990s.
Intelligence and socioeconomic status are certainly correlated, but I think it would actually be more correct to say “socioeconomic status is a proxy for intelligence” than the other way around. If intelligence is mostly genetic, with a small environmental factor, as it seems to be, and more intelligent people make more money on average, which they do, then socioeconomic status is certainly largely influenced by intelligence, and the heritability of intelligence at least partially explains why social mobility isn’t higher than it is. It’s probably the case that there’s some intelligence benefit from environmental factors resulting from socioeconomic status, but you can’t just try to throw out all the differences correlated with socioeconomic status, because there’s certainly some (likely more) influence in the other direction.
If measured average test score differences exist between genetic groups of people (they do), these differences again have to be some amount genetic in origin and some amount environmental. The amounts for between-group variation are probably similar to the amounts for within-group variation, but there are reasons why they might not be exactly the same. People often claim the tests are biased; what you can do to rule this out is to look at individual questions that certain groups of people get wrong more often, since it’s more likely that individual items are biased than that every question is biased by exactly the same amount. You can also look at things the tests predict, and see if they have the same predictive power across groups. For example, if someone claims the SAT is biased against racial minorities, you can look at college students in different racial groups and their SAT scores and see if the students in different groups with different average test scores do better or worse in college classes (I don’t know a good source off the top of my head, but to the best of my knowledge, the SAT is equally predictive across racial groups).
I strongly agree that the people claiming that psychometrics researchers have a racist agenda to prove white (or Asian?) superiority have a political agenda themselves, whereas the research in question seems to me pretty statistical, honest, and carried out without regard to the conclusions, and the researchers seem like they have to be very brave to carry out (or at least publicize) this research in the current political climate.