So before I answer, a couple disclaimers:
1) I'm Argentinian. I can comment some on what things are like here, but I only have a very basic understanding of what goes on in other countries
2) History was my worst subject in school, and I'm nowhere near an expert on anything involving social issues.
That being said!
First of all, are the terms mestizo and mulatto still in use? In English, the words used to describe nonwhite people 500 years ago are pretty much universally considered to be horrendously racist.
People are more likely to consider them outdated than horribly racist, but they don't get a lot of common use. Nowadays. mestizo means "mixed race" in general rather than any specific mixture of ethnicities, and sees some use. Mulato is probably more offensive, depending on where you use it. (also, the mulatto with two t's spelling is correct in English, but in Spanish it has only one t.)
Americans often treat "Hispanic" or "Latino" as terms to describe a racial group, which is clearly incorrect, and I think a lot of people here don't realize that there are black and white people who are Hispanic. Obviously, this is wrong, but I'm curious how people from Latin America think about this.
In my experience people from Latin America don't really have a term for their own racial group. If we move somewhere else we might identify as Latino or Hispanic, but not to ourselves.
Also, I think Latino is more used by Central America/Mexico than South America. At least, that's the local stereotype of people who use the term.
Second, what are race relations like in Latin America? I gather that, as is the case in the United States, white/fair-skinned people have significant privilege, but what form does this take? Overt racism in the form of segregation or an Apartheid-like system? Institutional racism, wherein laws and cultral expectations place darker-skinned people at a disadvantage? Economic inequality, wherein fair-skinned people control a disproportional amount of wealth, thereby making it almost impossible for others to escape poverty?
This varies a lot, actually. For example, currently Bolivia is having a strong pro-indigenous movement from the government, mostly led by President Evo Morales who is ethnically Aymará.
Racial relations in Brazil are probably atypical, since they actually have people with African ancestry whereas everywhere else the distinctions are mostly about how European vs Indigenous someone is.
But I'm gonna talk for a bit about the one case I know something about, which is Argentina.
A substantial part of the Argentinian middle and upper classes are of entirely European ancestry, mostly from Spain and Italy, which came here roughly at the end of the 19th/ beginning of the 20th century. This is less true the more north you move, where influences from Paraguay and Bolivia are felt more, and more so in the south. Patagonia in particular had most of its native population killed in genocidal campaigns and was then repopulated with the aforementioned European immigrants.
I did specify the upper and middle classes, though. Buenos Aires has a lot of people who move there looking for a job, either from neighbouring countries or from the impoverished parts of the north, which means that people who have noticeable indigenous ancestry are also more likely to be poor. This results in about what you'd expect, cultural mistrust, poverty traps, people crossing the street if they see a dark skinned person at night.
And yet, the thing is that it's not exactly a skin-based distinction. Skin colour is a continuum, so it's hard to say who is or isn't discriminated against by their skin, and usually a lot of class markers matter. A clearly dark-skinned person who can successfully signal being middle class is probably a lot better off than someone lighter skinned with a certain haircut or mode of dress or speech that marks them as lower-class. Plus there's a lot of non-skin related signals of ethnicity, and looking African or Caribbean will get you off much more lightly than looking like you're from the north or Paraguay.
Also, I'm not sure where to mention this but it's worth bringing up, the few indigenous communities left (e.g. the Qom) are typically treated like shit.
Thank you to anyone who can and is willing to answer any part of any of these questions, or direct me to a reliable source of information.
You did mention wanting books, and again this is not my area, but have you read any by Latin American authors? In particular, lots of people I know swear by Galeano's
The Open Veins of Latin America, and it should be easy enough to find a translated version, but I think it has the same political focus you say you have enough of. I don't know of any specific books dealing with racial relationships