Oh boy. My area of expertise
I'm an administrator on Wikipedia and have been for a couple of years. It's really hard to explain all the ins and outs of it, so if you have more questions you can ask or you can just check out
http://en.wikipedia.org/wiki/Wikipedia: ... objections
The system evolves and largely polices itself. Let me give you some specific examples. People want to vandalize articles, and they want it to be seen, so what do they vandalize? Articles about famous politicians, current events, etc. Of course, because these pages are so high profile, editors will have them on their watchlists and other editors will view them as well, meaning the vandalism will be cleaned up faster.
Regarding reputability and fact checking, Wikipedia has really cracked down on getting people to cite sources. If you go into any article of reasonable length or about an important subject, you're going to see tons of citations from books, papers, etc. I teach at a university and I let my students use Wikipedia as a source. Of course, they're just writing papers for class and not for a journal or anything. I think there are also studies by magazines (I think one was Nature) that showed that Wikipedia was more accurate and more thorough than Encyclopedia Britannica about various subjects.
About my involvement in Wikipedia... I started editing 3 years ago when I was working on my Masters thesis. I was studying some pretty esoteric crypto stuff and wrote some pages up about it. Later that year I was made an administrator. At that time there were something like 300 administrators of the English Wikipedia (about 600,000 registered users and millions of anonymous contributors). I wrote most of the articles on poker players there. I did a lot of vandalism patrolling, which basically consists of looking at every single edit that all the users make and then fixing it if it's wrong. Well, last summer I just couldn't keep up with it. A couple of edits a second I can handle, but it was growing too fast. So, I wrote a program that's commonly used by administrators there to catch and filter vandalism, called . It's pretty cool and I won't get into the details, but I won a prize at a computer engineering competition for it. Now that poker has taken over my life, someone else has taken over development of the project.

A friend of mine runs PlanetMath and I also started a content importing project at . My user page is and articles I've worked on are .
Anyway, a little more regarding use in classrooms. Various universities and high schools do use Wikipedia for projects and you can find some of them listed here:
http://en.wikipedia.org/wiki/Wikipedia:School_projects . Sometimes projects may involve something as simple as citing Wikipedia or doing research on it or may be as complex as creating a group of pages about a subject that Wikipedia is lacking.