Why is pubic hair coarser than the hair on ones head for example. (I have no research to back this up, but I think it's generally true)

no one knows for sure but the presumption is they evolved for different purposes. Head hair for warmth and thermoregulation. Pubic hair - not so easy possibly protecting the genitals from sharp objects (eg thorns).

This is an answer that I have never had reason to question/follow up on, so it might be slightly peculiar.  However, my old biology teacher told us that pubic hair might be used (primitively) in humans to signal sexual receptiveness.  Sweat (and the associated odours) evaporates poorly from skin, but more readily from hair.  By having pubic hair in the groin and armpits, the 'sweatiest' places in many people, our odours permeate the environment more effectively and they are picked up more by other individuals.  This is less desirable today, as 'we' as a species prefer people that smell like the average ground floor of a department store.

The follicle that a hair grows out of as a pubic hair is oval, not round like that on your head.  This means that pubic hair in most people is quite curly (unless you fall into a population with a round follicle), while head hair (may be curly, but) is not so tightly curled, and often straight.  Now bear with me, as this is where it might get a little crazy.  If the first idea stands, and pubic hair enable evaporation of sweat, having a curly pad, rather than a large flat surface, would be a better evaporative surface?  I know that this answer has a lot of speculation n it, but it is not a completely unreasonable hypothesis