1) I cannot offer a first hand POC opinion, but I can say that Yale has various cultural centers that graduate/professional students are welcome to be part of. Many YSPH students do take part in them and, although I do not have personal experience with them, my peers do talk about them positively. Additionally, YSPH has a diversity committee, and peers in my class have been very active with it this year, increasing its impact on campus. I believe Yale should have sent out a list of Student Ambassadors to admitted students by now. All students on that list are willing to respond to emails from admitted students. I highly recommend looking through that list for students involved in diversity or cultural groups on campus (which may be listed under Yale Affiliated Membership). Definitely feel free to reach out to anybody on the student ambassador list for a better answer!
2) Faculty support at Yale has been great. I will say that class size has been much bigger than expected. Because YSPH is a very small school, only one section of each class is offered. That means each class is as big as the demand. For example, your core classes will likely have your entire incoming class in them. Elective classes are much smaller. With this being said, even in large classes, faculty is very available. They hold their own office hours (in addition to office hours held by teaching fellows), and they are willing to meet with you outside of office hours by appointment. Faculty members readily take on MPH students for research, and in my experience, have been happy to meet to discuss topics even if they are outside of class materials. I do believe that faculty cares about students.
3. For specific international placement questions, I again recommend looking through that student ambassador list for students in the global health concentration or who list an interest in global health, as they are more likely to know about international placement more specifically. Second year students may have insight on the job search. All global health concentration students are likely to be doing their required summer practicum/internship abroad, so both first and second year students should be able to comment on that process. Overall, Yale's job placement is incredibly high shortly after graduation. We have a great career services office that is very approachable and willing to help students. YSPH students have a mandatory career services / professional development seminar course that you are required to take during your first semester in small groups, allowing all students to get to know and build connections with career services at the very beginning of your time at YSPH.
In terms of admitted students days, I know that Yale holds theirs on Mondays, which may allow you to attend Columbia's and Yale's in the same few days, assuming Columbia's is not on a Monday. There is a relatively inexpensive train called MetroNorth that goes from New Haven to Grand Central Station in New York.
I am an EMD student at Yale (who also considered infectious disease epi at Columbia), so feel free to ask me any program specific questions as well!