You assume financial aid puts students on par with those that do not need it. If you did go to college then you'll know that anyone working in the dorms or an on campus food court is a student who was on financial aid. And they worked those jobs because financial aid was not enough. Students that receive enough financial aid to avoid the need to work a job are the exception, not the norm.
Financial aid helps upward mobility but does not guarantee it. To think otherwise is pretty naive.
Yes and no. Obviously there are some rich kids whose parents give them spending money, and don't need to work at all during school. But plenty of parents, even from wealthy families, expect their kids to buy their own booze/gas/concert tickets/etc., and so, at least at my undergrad school (Williams), plenty of upper-class kids held work-study jobs and there was no stigma to it. I'm sure this varies by school, though. (as does the surrounding environment of what you can spend money on: big-city urban schools provide lots of opportunities for conspicuous consumption, whereas at small-town rural LACs like Williams everyone is equalized to some degree by living in the same dorms, eating in the same dining halls, and going to the same parties because there are no other options, so your work-study money really does go pretty far).