I'm in the opposite boat - an American applying to UK unis for postgrad - so I don't have much experience across the pond but I'm quite familiar with how the US university system works. Whether or not you're encouraged to join clubs really depends on the school you go to. I went to one with a strong sense of spirit and community (the University of North Carolina at Chapel Hill) and another where everyone did their own thing and mainly focused on internships and other career-related activities (New York University). Both are great schools, and I'd recommend either one. For what you're looking for (I'm guessing lad culture is like our fraternity system? Like a bunch of guys get together and drink and socialize?), UNC Chapel Hill would be a great option. Other well-ranked public universities in college towns would also offer a similar experience. UNC has a lot of greenery, fraternities, sports (soccer/football isn't big here - just basketball and American football), and strong programs in pretty much every area (including business, comp sci, and psych). I'd definitely recommend it as an option.
At US universities you start as an undecided student (meaning you aren't focused on a specific subject), and the university mandates that you take subjects in a wide variety of areas. The mini courses you referred to are electives, and they're basically classes in areas other than the subject you want to focus most of your studies on. After a while (generally by the end of your second year), you pick a major (you can also do this earlier), then you focus a lot of time on taking classes in that subject. Many students double major, meaning they pick two subjects to focus on. You can also minor, meaning you take a few courses in a subject (but not nearly as many as you'd take for your major).
You'll want to go to a well-ranked school or a smaller private school (like Swarthmore) if you want passionate students who aren't just there to tick a box. Schools like the Ivy schools are really competitive and I'd honestly say a lot of those students are probably there to tick off boxes.
SATs are standardized tests that cover reading, writing, and math. There's a heavy emphasis on vocabulary, so people stereotypically have to memorize a bunch of vocab words to do well on the test. They aren't like GCSEs or A levels because they aren't part of your regular school classes. They're a separate test that you study for and take outside of school, and you don't prepare for them in your classes (some teachers teach vocab words, but there's no requirement to do so). Some people pay for practice SATs or take courses to help improve their scores, while a lot of other people don't. They're long tests - like 4 hours long if I remember correctly. Students generally take standardized tests in 11th grade (their second to last year before college) and early in 12th grade (their last year). They pay a fee to send their scores to universities, who use the scores to help inform their admissions decisions. Most well ranked schools judge applications holistically, meaning that you can get into a school with poor SAT scores. There's also the ACT, which is a similar test. You may be able to take the SAT at a UK testing center, although I'm not too sure about that.
Hope that helps! Feel free to ask if you have any other questions.