On Rank, I would consider the relative position of people working in your subfield at the institution under consideration, not the department's rank overall. Some departments have incredibly strong faculty in one field (IR, CP, American, Theory etc) but not in others.
On Academic Fit, I would consider whether your epistemological outlook is aligned with potential PIs. Substantive topics aside, are people doing the kind of research you want to be doing (natural experiments, ethnography etc). Will advisors allow you to do the work you want to do, the way you want to do it or will they push you to become more like them (this you can get a good sense of from talking to them about your work, but also current graduate students).
On Advising, I think you covered the bases. I would also consider whether your potential PI has an advising style that suits you. Some people are very hands off, and that doesn't work for everyone but this fits with engagement. It's worth it to ask current and previous students what their meeting schedule looked like, and whether they felt their advisors read the papers they send and offer useful feedback. You don't need a committee of three people who will read everything with a fine toothed comb, but it helps to have one. Also, ask yourself if your main advisor left, would you still want to go to that school. Faculty move a lot, and most are not in a position to bring students with them or continue chairing committees if they leave and you can't follow them. I know too many people to count who have lost their primary advisor to a move, and then felt stuck committee wise without their main mentor. This is somewhat related to the idea of not choosing a program to work with one specific person.
On the Cohort dynamic - ask about office space. Do places have it for graduate students; do you have to compete for it; is it a positive or negative work environment? This seems pedantic, but it can mean a lot for positive social and academic environments. It changed my grad school experience drastically when we got access to a building where all graduate students have dedicated offices (if they aren't working out of specific centres). I have two potential co-authored papers I doubt I would have in the mix if I wasn't working in such an environment. Your immediate cohort will only matter for the one to two years you are doing a lot of coursework, so it's worth it to consider the general climate amongst graduate students, and whether people are hostile or constructive in feedback and collaborative opportunities.
On money: I would add a few other points of consideration.
What is the cost of living in immediate area? Is rent so expensive it takes up your entire stipend? Will you need to commute to make ends meet if you can't get on campus housing? Is commuting easy (reliable transit, 15 min drive with no traffic) or difficult (no transit, heavy traffic/long drive)?
Is your stipend the same as everyone else? I.e. do students compete for better funding packages. This is surprisingly true for a number of programs, and it can generate hostility in cohorts if people are fighting for money.
Is there accessible funding for sixth years? Do you have it guaranteed or is it competitive?
Is there accessible funding for fieldwork or research protocols? I.e. how easy is it to ask for 5000 to run a survey or spend a month in an archive? Are those internal departmental options, NSF grants, research centres?
For RA work, what is the typical wage? Are RA/TA obligations built into your stipend or are they an addition to your stipend (i.e. if you TA for a semester, do you earn additional wages on top of your stipend or not - this varies a ton!)
Are you responsible for any annual fees or health insurance, or are these included in your funding package?
On location, I would also ask if it's a place you can see yourself living for at least 3 years, if not the entire duration of your program. Can you lead the life you want in the area the school is, with what kinds of housing is available to you etc? This is important for pet owners, as well as people looking to settle down or find a long term relationship in grad school. It's also true for people moving with partners or children. If the immediate location doesn't work, but a neighbouring city does, consider whether the commute is something you can handle (and afford).
On partner groups, I would also consider whether centres regularly bring in post-docs or visiting professors. It's a great opportunity to collaborate with early career scholars, and bring in additional expertise. Minor thing, really, but it adds to the climate of collaboration and opportunities. Also consider whether there are research and social groups that can support you as well. I've gotten a lot out of a women and politics group we have in our department as it has fostered connections between female graduate students and female faculty I wouldn't normally interact with. Likewise for first generation scholars, or visible minorities, some departments have a lot of great opportunities to network with peers that can make the PhD process more manageable.
A lot of people told me that such questions weren't important. A grad student at a visit laughed when I asked about offices and said that surely I wouldn't pick a place based on whether I would have an office or not. Sure, my decision wouldn't hinge on that one factor but it's a question meant to probe the underlying social dynamics of a department that may not be immediately visible during a visit weekend. There is plenty to consider of course. Hope this helps.