There's a bunch here, so let's dig right in. I'm sorry, but this is going to get a bit complicated because the problems with Zinn are much more involved than his just having an agenda.

In my AP US History class we have to read Zinn's People's History, which I have never read, but I've have heard from numerous reliable sources that Zinn is a bit of a hack. If I am understanding correctly, he intentionally wrote a piece that emphasizes his oppressor-oppressed narrative, or something like that?

Sort of. The history of marginalized and oppressed people is a vital part of history and should not be discounted since the alternative is basically to insist, at least in the context of US history, that only white men matter even whilst discussing them committing major atrocities. It's easy to slip into doing that because our culture tells us the same thing, so we have to make a special effort to unlearn those assumptions. It feels unnatural and agenda-pushing because it doesn't square with the agenda that's been given to us by the wider society essentially from our births, whether we had someone tell us that only white people deserve to be considered as people or not. So far as Zinn is doing that, he's not doing anything particularly wrong or hacky.

Whether People's History is a good example of that or not is a larger question that I need to defer since I haven't read it in this century. But I can look at the passages you've quoted.

In the history of the world, there is no country where racism has been more important than in the United States.

This might be true, depending on how one defines racism and measures its importance but it's not actually a useful observation. Zinn fails to tell us how racism is important in the US or why it's so unique. That doesn't give us anything to work with from a historical perspective. A better observation would be that white supremacy has been central to American society, both through the genocide of Native Americans and the enslavement, persecution, and oppression of black Americans from about 1621 onward. Zinn could also have pointed out that the color line was especially rigid in the US compared to other New World slave societies or that the United States was home to the world's only fully-developed school of proslavery theory.

Could someone explain this general idea in more detail, about how the colonists were not only economically but psychologically suited to be cruel slave owners?

Now that's a complicated question. Morgan's work, which would have been a better APUSH reading than Zinn, is remarkable in that it's an important work still taught coming up on fifty years later in a pretty busy field because it's so good and generally considered dead on. To the best of my recollection, Morgan doesn't stress envy or anger as making Englishmen in Virginia psychologically ideal enslavers and subsequent work hasn't picked up that argument much. So far as I can tell, it's Zinn editorializing without evidence. The present scholarship on the origins of white supremacy is vast and muddled in various details, but actually tends to stress that at first Englishmen were not singularly vile toward black people. They were still pretty bad, but they were terrible to others they deemed inferior in general and black men who converted to Christianity and won their freedom seem to have been treated at least roughly equivalent to whites of the same status. Probably not exactly the same, but we have evidence of one who did well enough that he was able to sue white neighbors in court and win, ended up a substantial property owner, and was well enough accepted that he even got a few years abatement on his taxes after suffering some misfortunes. Rather the story of Colonial America is more one of white supremacy developing than suddenly erupting out of envy and anger toward Native Americans.

American slavery was the most cruel form of slavery in history because of two things. First, American slavery was driven by a frenzy for limitless profit. Second, it was based on racial hatred, a view that saw whites as masters and blacks as slaves. For these reasons, American slavery treated slaves as less than human.

This is a frustrating mix of right and wrong. American slavery was profit-driven, but that doesn't make it unique. Nobody went into slavery to not make money or otherwise get something out of it. Likewise it was based on racial hatred, which does distinguish it from some prior slaveries but not from anywhere else in the New World. Furthermore, modern slavery studies hold that slavery of every kind is, in essence, a practice of treating slaves as less than human. The simple fact that it's slavery makes it dehumanizing and no enslaved person, in any place or era that we've yet studied, has been treated as fully human. Zinn's explanations of that fact are true of the American situation -enslavers want profit and dehumanize those they enslave- but that's true of slavery everywhere. Setting the US apart here is like trying to find water that doesn't have any hydrogen and oxygen atoms.

African civiliza­tion was in some ways more advanced than that of Europe.

For the most part, historians don't make these kinds of comparisons anymore. Advancement has long been a measure we use for how human we treat others. It is certainly true that the West African coast that enslavers got their human cargo from had states, agriculture, iron tools, cities, and other things that we could say make them "civilized" but even if they did not that would not make the people who lived there in any way lesser to those who do have such things. What Zinn assumes, whether he realizes it or not, is that it would somehow be more legitimate to mistreat people who lived in cultures without cities, states, and so forth. He's trying to argue that Europeans were hypocritical as well as factually and morally wrong to enslave Africans on the basis of their (Europeans) being a more advanced people, but playing into the advancement dynamic gets you right back where they started. Had he simply argued that slavery was a brutal, dehumanizing system to which no one would have been subjected, he would have been fine.

Furthermore, the Europeans who first bought enslaved African people knew exactly what their civilizations were like. They went down on ships into their ports, dealt with their leaders, and were forced to accept bargains they didn't always like because African polities were, and would largely remain, too powerful for Europeans to force things to go their way. So Zinn is trying to explain slavery as though the actual enslavers are ignorant of things literally in front of their faces.

Only one fear in the American colonies was greater than the fear of black rebellion. That was the fear that whites who were unhappy with the state of things might join with blacks to overthrow the social order.

This is possibly true, but the devil is in the details. Solidarity between poor whites and blacks of any class was a concern of white elites and they do take steps to protect themselves against it. These largely entail codifying white skin as the marker of freedom and black skin that of slavery, so that any white man can look down upon every black man. This is sometimes presented as a con job by the rich dudes, but I think that's going quite a bit too far. Poor whites were not dupes. They got a deal which was not ideal for them and definitely not actual equality among whites, but they did receive genuine advantages that helped them buy into the system and the cultural differences that whites come to think of as marking themselves as superior are part of a heritage shared by rich and poor whites from the get-go. You wouldn't need to convince a 16th century Christian Englishman that Christianity was superior to other religions and that non-Christians didn't deserve the same kind of consideration; he's got that down already. Likewise he's going to think the clothing styles, marriage patterns, and so forth that he grew up with are natural and normal while others are weird and maybe mean something is wrong with those who practice them. All that being the case, the fear of interracial solidarity is likely overblown in most cases and that becomes consistently more true as Virginia's first century rolls on.

So what's the takeaway here? It's fine to read Zinn -and you've got him assigned so you need to anyway- but you do need to read him very critically. We all have agendas and we are all always pushing them. They inform everything we do. That's true of all history texts, of course, but it's a whole lot more true of his work than that of more recent and more rigorous historians.