Earlier quoted context omitted.
The reason I've been reading it all is because I'm writing my thesis on it :) I haven't spent too much time with Khan Academy, so can't say too much about it. There are two things to keep in mind: 1. Extrinsic motivation erodes intrinsic motivation. The most obvious example is asking a kid to take out the trash. Once you pay him/her, they'll only do it for money again. While this example is silly because no-one has a…
For me, Stack Overflow's most rewarding feedback is social recognition for having been helpful, which is largely independent of the formal rewards it offers. I bet it's the same for a lot of other people too.
The literature on motivation from people that subscribe to self-determination theory indicates I think they would mostly prefer the system gone. Any reward system you put in place is bad, it's just there are certain degrees of bad.
My gut tells me that Stack Overflow has benefitted from the reputation system as another explicit feedback on how well you are doing. I think the badges are largely unimportant, save for being a more fun way of saying "here's what you can do on the site" than another tedious FAQ. If the badges were important, then we head down a nasty extrinsic motivation road. I think it's done better with the rep system than without.
That said, I sometimes wonder how many people are put off by the built-in competitive aspect of Stack Overflow; any writing you do is evaluated and measured against others. For competitive people that end up bumped down all the time, that seems like it would be disheartening, for social people always been graded it might be disheartening too. The winner-takes-all set up, rather than, say, a wiki answer that people are constantly tweaking, seems like it could be problematic. I know that I stopped bothering answering questions very quickly as I felt I had no mastery of the art, other people were getting the upvotes, so I felt like my answers were largely ignored.