As for your question, g is very difficult to improve through training. Almost impossible in adulthood. The only studies I am aware of which indicate an ability to improve IQ score over time are those which allow participants to repeatedly take similar IQ tests. This enables the respondents to practise the type of question and answer faster and more accurately. Obviously this would produce an inaccurate result, over-estimating the intelligence of the individual.
To draw parallels with the SAT, I see evidence that repeated attempts reveal gaps in one’s knowledge and skill, and these are prerequisites for the test. This permits the individual to study those areas and practise the kinds of skills required to improve their score. Given this, we should acknowledge that the SAT is not a perfect analogue for IQ tests, but in aggregate, they do yield striking correlation. To tie this to my premise above, it is highly likely that the average Harvard student has an above-average IQ.