re #5: “I’m mostly looking into pure mathematics and/or computer science.”
Regarding pure mathematics maybe some of these prior threads can help:
http://talk.qa.collegeconfidential.com/harvard-university/223626-if-not-harvard-where-else-for-math.html
http://talk.qa.collegeconfidential.com/princeton-university/900687-cambridge-vs-princeton-for-math.html
http://talk.qa.collegeconfidential.com/college-search-selection/1319347-math-major-harvard-stanford-princeton-caltech-or-berkeley-p1.html
Regarding computer science, maye some of these prior threads can help:
http://talk.qa.collegeconfidential.com/college-search-selection/932455-the-best-computer-science-undergraduate-programs-p1.html
http://talk.qa.collegeconfidential.com/math-computer-science-majors/1943448-top-10-computer-science-universities-p1.html
also these:
https://www.topuniversities.com/university-rankings/university-subject-rankings/2014/computer-science-information-systems#sorting=rank+region=+country=+faculty=+stars=false+search=
https://www.usnews.com/best-graduate-schools/top-science-schools/computer-science-rankings
http://www.shanghairanking.com/SubjectCS2012.html