Is general intelligence meant to underlie all intelligence? Is general intelligence, supposedly measured by IQ, meant to underlie all intelligence, be the thing that accounts for all intelligence / ability, or does all intelligence account for general intelligence? Who claims the former, that g is more basic, and why do they claim that?

I think the question matters a little, because, if it doesn't, then IQ scores are a product of learning and teaching successes, rather than vice versa -- given that ability in any sphere [mathematics e.g.] is not innate.
Is general intelligence meant to underlie all intelligence?


I don't know what you are trying to ask.  Are you trying to ask if general intelligence is the basis of all other intelligence?  If this is what you are indeed asking, the answer is no.  Long ago, scientists thought of g, general intelligence to be a number that encapsulates the sum of all of one's different intellectual abilities.  (I know.  It sounds vague and doesn't sound very scientific.)  I don't think they intended to mean that g was like some precursor to each type of intellectual process and thus, the best indicator of a person's intelligence.


  Is g, measured by IQ, the basis of a person's intellectual abilities
  or does a person's overall intelligence lead to a number, g?  Who
  claims the former and why?


I reworded this person's question because it wasn't very clear as written.  I also think the question is almost exactly the same as the first question, except the person also asks who believes in g as the basis of intelligence.  

There was once a time that most scientists and other experts in the field of intelligence believed in the concept of g, a number that encapsulates all the intellectual abilities of that person into one simple score.

As it still happens frequently today, in the early 1990's, there was a remarkable discrepancy in what most scientists believed intelligence was and what popular news articles and media such as movies believed.  (This is a topic for a different question but it amazes me sometimes the huge difference between what scientists believe and what the public believes.  I personally don't think this vast difference just happens by accident.  The information conveyed to the public is usually distorted intentionally.)

Anyway, going back.  In 1994, scientists in the field of intelligence were shocked by the public's completely backward notions of what intelligence was.  The media was apparently butchering what scientists were saying.  As a result, some 50 leading scientists and experts in the intelligence community signed a list of statements that was published in the Wall Street Journal, expressing dismay at the media's incorrect depiction of intelligence.  This public statement was titled "Mainstream Science on Intelligence".  It stated facts that were widely accepted by scientists and experts in the study of intelligence.  Some of these statements were:


IQ can be measured well and IQ tests are very accurate.  
There are different types of intelligence tests but there is only one type of intelligence.
Intelligence tests are not culturally biased.


Such outrage by the scientific community!  What a shock!  Why would the media intentionally distort what the real science was?

Well, all these questions became moot because something really ironic occurred. 
 It was determined that most of the claims made by Mainstream Science on Intelligence were wrong!  It's ironic but no reputable scientist or expert in intelligence today would ever sign that statement that they had once so confidently signed and published in the Wall Street Journal.  Studies made during and after the 1990's, actually disproved most of the statements in Mainstream Science on Intelligence.  Scientists did a complete 180-degree turn.  The lying media turned out to be right!

What's more ironic is that there are now laypeople who believe in the statements made by Mainstream Science on Intelligence.  Not scientists but laypeople who haven't been keeping up with the science.  They are accepting the statements made in the Mainstream Science on Intelligence as true.  They believe things like IQ is an accurate and precise measure of general intelligence, g.  Or, g actually is a great indicator of how smart a person generally is.  Funny, no?

Anyway, the original question had asked who claims that general intelligence underlies all intelligence.  I don't think anyone believes this.  I never heard of anyone who believes that there are some group of genes (aka g) that has an effect on all our intellectual abilities and is the factor that mainly accounts for a person's intelligence.  Next question...


  I think the question matters a little, because, if it doesn't, then IQ
  scores are a product of learning and teaching successes, rather than
  vice versa -- given that ability in any sphere [mathematics e.g.] is
  not innate.


I didn't understand this question.  How did you logically conclude "IQ scores [are] a product of learning and teaching successes" since "IQ scores [are] incorrect representation of intelligence"?  It happens to be true but how did you conclude that from anything you said before or after?  And, what do you mean by vice-versa?  It doesn't make sense.  

To set the record straight, there is no such thing as a g that represents the general intelligence of a person.  Just because you are good at one thing, it's been determined that you might not be necessarily good at other intellectual pursuits.  I strongly believe this to be true.  Research has proven it.  And, you personally see instances of this in real life.

Let me also state that an IQ test is also a very inaccurate method to measure intelligence, especially at the tail-end of the distribution on the right side.  Furthermore, please keep in mind that the scientific field of intelligence is actually still in its infancy.  Many basic concepts are still unproven to be true or not.  We are still learning about ourselves.  Current studies include determining what types of events appear in what part of the brain.  It's all so very new.