For private universities, there is no difference in fees for an international student and an American student. For state universities, there is a difference in fees between in-state and out of state.
For an idea of the best colleges (universities with just undergraduates departments) and National Universities (includes graduate divisions), this is a very good resource. It lists fees, statistics on student life and financial aid.
Some schools have needs-blind admission and the student is accepted without any idea of what their financial situation is. If they are accepted, the school will assess how much the student and family can contribute and fill in the gap with a mix of grants and loans. To give some idea of financial aid available, Harvard University will give a full ride (tuition, board, travel stipend, living stipend) to those whose parents make less than £75,000. However, schools like Harvard, Yale, Princeton and other generous schools also are extremely academically/extracurricularly selective - acceptance rates of 4.8%- 6ish%.
In terms of selection, simply being good academically is not good enough. 3 years ago (the stats I know very well) over 700 applicant to Harvard had perfect scores on their SATs (not the equivalent to SATs here, but a basic college entrance exam) - of these, they only took about 200. They prefer a student with a lower score, but brings something else (very significant extracurriculars) to the table. The admissions office said that they were not looking for just heads on sticks, for a dynamic person. Student activism, meaningful volunteering experience demonstrating the applicant made a real difference, or a very unusual accomplishment are really favoured. This definitely does not mean being sporty - apart from being nationally ranked.
The academic requirements would be to take the SATs (not difficult at all - maths, reasoning and comprehension, and a short essay - preparation books can be purchased on-line), a school transcript, 2 letters of recommendations, as well as the application (which requires 2 essays usually). Many of the top schools are familiar with the academic qualifications in the UK and have quite active outreach programs (DH is very involved in one, which is why I have loads of info).
In terms of accommodations, most students at the top schools live on campus in dormatories or fraternity/sorority houses. There is a huge emphasis on campus life at these schools.
I could be wrong, but I always thought that Fulbrights covered graduate studies, not undergraduate.