I've been on the grad admissions committee at my university for 30 years, and have been Dir of Grad Studies in math on two different occasions, so I have a good sample of what goes on here.
First, in the last few years all our applications are electronic, so the admissions cte doesn't have to be in a little room with piles of paper any more. This also allows much more asynchronous appraisal of files, which allows not wasting time trying to figure out common meeting times.
Typically, a preliminary screening is done in a "distributed" sense, that is, obviously-not-qualified or misguided applicants are removed from the "active possibilities" queue. By this, I mean engineering or computer science students, or crackpots, who declare interest but have no documentable mathematics background, no letters of recommendation from mathematicians, no GRE subject test, no nothing. :)
Even after this initial filtering, we have many fewer (funded as TAs or RAs) spots than approximately-qualified candidates. Even though we expect maybe 50-percent accept rate of our offers, the number of approximately-qualified candidates is still too large by a factor of 2 or 3 or more.
Candidates from small colleges in the U.S. typically have a very thin background, whether or not they've done special summer programs, so are at some initial disadvantage in comparison to candidates from Europe or Asia who have seen much more coursework, if only due to the structure of "college" there. This also precipitates discrepancies in GRE subject test scores. Nevertheless, we have found that this discrepancy is often very temporary, and after two years, or less, success in our program is almost completely uncorrelated with such things. Thus, many of the pretend-objective measures of prior success are of very limited value.
Further, the import of letters of recommendation depends greatly on the author, and their prior experiences, basis for comparison.
So we are left trying to compare candidates from widely varying backgrounds...
To get to the "final list" of offers thus consists partly of winnowing out candidates who are perhaps vaguely qualified but indicate no genuine interest in anything about our program, while putting onto the "definite" list candidates who seem to be focused and have a particular interest in the research program(s) of our faculty.
Documentably strong background and prior success is good, obviously, but, again, undergrad work, summer REUs, and such are a different enterprise than genuine graduate work and long-term research. Success in a highly structured environment gives very ambiguous indication about success in an unstructured environment. Success when there are unambiguous answers to questions that can be answered in a few hours is different from success in situations where the program can take months, and one is not quite sure what the question is.
So there is considerable discussion and re-reading of personal statements and letters of recommendation, trying to project into the future and a new environment, based on our prior experience. Not easily turned into an algorithm.
As to the specific questions: an obviously implausible applicant could be appraised in 5-10 minutes. At the other end, the final decisions about the better obviously-qualified applicants may involve repeated appraisal, so that the total time spent might be 5-10 person-hours in the end.
As to whether we look at on-line manifestations of applicants: this is a potentially delicate issue. The default approach would be to absolutely not do so, certainly not look at Facebook or other social media. If the application gives a pointer to a "professional" web page, of course this is considered useful, and positive if that site gives evidence of good work of the applicant. But "snooping around" would be unethical, inappropriate. Even if there is a pointer to a site in the application, if the application is not otherwise plausible I don't think I'd look at the web site.
EDIT (29 Aug 2020): since this thread got bumped to the top anyway, I thought it'd be not too disruptive to update remarks about GRE subject tests: the ETS (Educational Testing Service), which creates and administers the GRE, have cancelled the subject tests for this autumn (in the northern hemisphere, I hasten to add, having someone recently note my hemisphere-ism! :)... As I've said in this thread and others, in fact my dept does definitely does not consider GRE subject test scores good predictors of future success. The only aspect of interest has always been whether or not students succeed in understanding that (in the past) many or most places did consider a subject test score essential to an application... so we here would wonder why a student wouldn't have that score, even if we didn't care what it was...
Now, the Dir Grad Studies office is composing a highly revised statement of our attitude about the GRE subject test, at least saying that we are not interested in it, do not require it, and do not want to see it. (Apparently my math dept does not want to make any remarks that might be interpreted as "snarky"...)
