Hey Joniman, I’m a current undergrad that is in a traditional fraternity here at USC so I’d be happy to answer your questions.
Firstly let met address the fact that USC is an extremely, extremely diverse and quite liberal university with a huge undergraduate population… in the middle of Los Angeles. You’re not going to find a more diverse melting pot of culture in many other universities like here. While there are some underlying problems with racism at the university (like most universities in the US, mostly towards international students from Asia who don’t speak English as a native language), it is very subtle and it is always being addressed and discussed, while cases are still fairly rare. (So rare that most of the time big cases end up on national news…)
Okay onto the questions…
- Simple answer: Yes. I'm sure most people have a preconceived idea of what a fraternity is, but the simple fact that having ~22 frats at an elite university means that we're a fairly cultured community of 'bros'. The Row is a very welcoming (but dying, as the school continuously tries to fade us out) community, although it may not seem like it to outsiders. While some frats have a more consistent history of having a dominant caucasian population, every house will accept anyone regardless of race or sexual orientation as long as they 'fit' in along with the rest of the guys. If it helps, one of my friends is African (not African-American) and he's in a frat too.
- I'm not going to recommend a frat that you have the most "chance" with for several reasons. Again, it sounds like you're going off preconceived notions here, but I'll tell you what you should do when looking for a frat at the end of this segment.
Firstly, it kinda doesn’t matter how wealthy you are as long as you can pay your dues (generally around $2000 - $3000 a semester) and, to be honestly, dress and act well. That being said, most people on the Row range from middle to upper class.
Secondly, the reputations of the frats change so often that anyone reading this response in the future will almost definitely have a different type of Row when they’re interested in joining.
But to address hazing… just be ready for anything. No matter what people tell you about a frat, even if they say they don’t haze, everyone who does haze hides it very well to the public eye. Honestly, almost every frat is going to make your pledge semester extremely taxing (and maybe even a living hell). That’s just how it is, but we all got through it somehow right? 
In terms of picking a frat, you’ll understand every frat’s reputation when you start studying here. I wouldn’t recommend checking out the reputations online at the moment just for the sake that we’re all kind of getting in a lot of trouble and houses are either getting banned or cut left right and center, plus the vocal people on the Row like to talk smack to each other, so most “critiques” (not rankings, but like the written reviews of a house) you’ll see online are usually the minority trying to hype or trash the house (it’s funny and sad at the same time). The best way to be informed on how the Row is doing in terms of reputations is to hear from it from people who have ACTUALLY BEEN ON THE ROW in the previous semester. It’s a lot of petty politics, so get ready for that. Also, everyone cares about academics, so don’t worry about that aspect of the frat (except when you pledge… that’s probably going to be a rough semester for you).
- Common in dating? Meh, it's equal to how it's going on in the rest of the world. Let's just say that nobody would care. Same goes for hookups. It all matters on how you are as a person, not the color of your skin.
- To be honest I notice that some people like to hang out with those who share a similar cultural background. It makes sense I guess; it makes it easier for people to get along with each other. But I personally (as a caucasian) do not even think about race when talking/hanging out with people. I feel like most people feel the same way around campus, so in terms of socially speaking, everyone basically just mixes together. But I still have to point out that caucasians ARE the dominant race on the Row so... we're welcome to any race joining but you're going to have to be okay hanging out with a lot of white guys haha.
One last thing. Remember that by rushing (trying to join) a traditional fraternity on the Row you are attempting to get into an exclusive club for guys (yeah it sounds stupid I know). While statistically speaking you have have a high chance of getting rejected, it all really depends on you. Looks, character, humor, charisma… everything. We all want the best guys for our houses, so naturally we’ll reject a lot of people who just want to go to parties but don’t actually fit into our type of social scene. Frats aren’t for everybody, and you shouldn’t join it if you aren’t comfortable being in outgoing situations on a constant basis.
Hopefully that answered a lot of your questions! If you or anyone else have any other questions about the Row, I’d be happy to go into even more depth, but for now I’ll leave it here.