Hi, I have a PhD in a different field.
- Every program does this differently. Some programs have a date on which they release all decisions, although they rarely transmit that information to the applicant (just in case something delays that date). Most programs have been doing this for a long time and have a good idea of their annual average yield, so they decide how many students to admit based on that expectation (i.e., if you know your annual average yield is around 70%, and your goal is 7 students, you admit 10.)
Another common way PhD programs manage yield is exactly what you said, by offering admission to their strongest applicants first, and then putting the other applicants on a wait list. Sometimes that wait list is formal (i.e., they let the applicant know that they are wait listed) and sometimes it is informal (they just leave the wait listed applicants hanging until they hear back from their strongest applicants).
- Usually, yes, they do come with the stipend and fellowship offers. He will be able to make comparisons, but they won’t exactly be apples to apples. First of all, housing offers rarely come with the acceptance - housing is usually secured after you’ve accepted an offer. I’d advise him to research housing now if that’s a concern, and he’ll have to make his decision based on estimates, really.
Second of all, most programs will tell you “expected” work load, but there’s often an element of prevarication 
In other words, the vast majority of programs will say “Our research assistantship is 20 hours a week.” The RAship is almost certainly not actually 20 hours a week (it’s more). What it actually is depends on the program and the PI, so I would encourage your son to talk to graduate students already in the program, and particularly those who already work for the PI he wants to work with, to find out what workload expectations look like.
- Aside from stipend amounts and location, in very rough order of importance:
-Prestige/reputation of the PI/placement rate of the PI. Where do PhD graduates end up after they finish? How often do students from the department get prestigious national fellowships like the NSF, NDSEG, and Hertz? (Those he can look up on their respective websites). This your son can find out by asking the PI themselves to give him an idea of where his recent graduates have ended up in the last 5-10 years. (He might have to reach back, since science PhDs often spend 3+ years in postdocs before getting an assistant professor position.) If he says he doesn’t know, that’s a red flag, since it’s his job to know.
-Prestige/reputation of the department/placement rates of the department. Ditto the above. Some departments have this on their website. If they don’t, call and ask. If they say they don’t know or don’t keep records, treat it as a red flag. It’s their job to know. (It also may be an indication that they have little to no professional/career development going on in the department.)
-Productiveness of the lab and the students that work in the lab. If the students have their CVs up on the website, check them out (they may be out of date, though). Same with the PI. Is the PI churning out papers and grants like a machine, or is the last thing he wrote a monograph from 1994? Scientific journal articles are the currency of the field. It’s how you get jobs, postdocs, grants, and other opportunities. Good PIs offer opportunities for students to get authorship early and often. Excellent labs have students publishing as early as their second year. If students don’t seem to be thinking about papers until their fourth year or later, that’s not a great sign, especially not in the physical sciences.
-Nature of the offer. Fellowship is the best level, because technically he doesn’t have to do anything for the money and has the most choice of who he can work with. Research assistantship is the next best, followed by teaching assistantship at dead last. Everyone needs a little teaching experience, but too much takes up time you could be spending on research (which, again, is how you get jobs and grants). Most offers are some kind of mix of teaching and research - most common is to require you to TA a class once a year or so. This is going to be tied to the above - the most productive departments are likely to be the ones that don’t require a metric ton of teaching.
-Supportiveness/nature of the PI he will be working with (or the general atmosphere of the department, if he is unsure about the PI). Is he a horrible boss that expects 100 hours a week of work and barely ever sees his students? (They exist!) Is he a “too nice” PI that gives weak feedback on drafts, doesn’t care how much you work and also almost barely ever sees his students? (They also exist! Often in the same department!) This can be gleaned from talking to current graduate students, mostly. Ask them how much they work, what their lives are like, what their PIs are like, etc. A good PI is somewhere in the middle - has high expectations of student work and gives supportive but critical feedback when necessary.
-Professional development opportunities in the department. Are there career workshops? What about academic job market ones? Is there a departmental pot to send students to conferences, and how big is it? (Or do you get an individual pot? That’s the best!) How often do students present their work at conferences? Do current students talk about hobnobbing with bigwigs in the field or do they seem to not really know what’s going on? Are there grant-writing workshops? Do students author grants with their PIs? Do they know professors in other departments? Talking to grad students again is the key here, and asking them about recent conference experiences and workshops. But chatting with the PI is another way. Ask him what kind of professional development opportunities he provides for his students. If he has no idea what you are talking about, yellow flag! If once you explain it he recovers and says something interesting…yellow flag at half-mast.
-Time-to-degree. This is going to be tied to the above. But how long does it take PhDs to finish? Are they speeding through in 4-5 years or are they languishing around for 6-7? You can ask the departmental secretary about this; she always knows. If she doesn’t, she can get it from the DGS. If nobody knows, treat that like a yellow flag. Departments that don’t keep track of this are often (but not always) the departments in which students take the longest to finish, and they don’t track it because they don’t want to know.
-Prestige of the university. If your son aspires to an academic position, this doesn’t really matter that much - it’s more about the department. But if he’s considering non-academic careers, it will matter a bit more. I’d leave it at the bottom of the list as an absolute tie-breaker. I kind of went to a prestigious university for my PhD on accident, but I think it had a moderate impact now that I’ve gone into non-academic research.
Different people have different opinions on where location should reside on the list. One argument I’ve made before is that going to the best program for you in an undesirable location may give you more flexibility after your PhD - you’ll be more competitive for the top departments in better locations. Another argument I’ve heard - that I also agree with - is that if academia is your goal, your graduate school might be the last chance you get to pick where you live. I think my conclusion at this point is that delayed gratification stinks and a PhD is still 5-6 years of some of the prime years of your life that you’ll never get back. If you have the chance to live somewhere awesome, take it. I have mixed feelings about having gone for a PhD, but one thing I don’t have mixed feelings about is getting to live in New York for 6 years in my 20s.
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