How to check the veracity of arXiv.org papers? Suppose I found a preprint on arXiv.org (like this one: http://arxiv.org/abs/math/0309146 ), and I need to check if what's in there is actually true. Any tips? I don't want to study an article only to later find that it contained an error <_<
More generally, should I spend my time on arXiv.org preprints? How serious is the danger that I will find myself holding a bunch of false beliefs and unable to publish my own works in any serious journal because I can't reference arXiv.org?
How do I find if an arXiv.org preprint was actually published in a peer-reviewed journal later? What is the average time interval between sending an article for peer review and it being published? If the paper was not published in, like, 5 years, is this a good reason not to trust it?
 A: The simplest way of finding out if an arXiv paper was published is to check the arXiv to see if they list a publishing date; many papers do (those of mine that have been subsequently published certainly do). Absent that, contact the author(s) and ask. (As Willie Wong notes, the paper you refer to has such a listing; that paper has appeared in print already; of course, you'll take into account the quality of the journal if this is the case). 
There is risk of a paper being wrong even if it has been published in a peer-reviewed journal, of course. The odds are lower (one hopes) than for papers on the arXiv simply because third parties have taken the time to go over the paper before publication. The same is true of papers in the arXiv, though less formally, so you'll want to check to see if anybody has read the arXiv paper (check in fora such as this, or sci.math.research, or in specialist mailing lists, such as the GroupPub Forum).
You may want to check the author's track record; and of course, you'll want to check the paper before you start quoting it (this is true of published material too!). 
As for trusting a paper that has not appeared, there are many possible reasons for a paper not to appear that have nothing to do with its correctness. The results may have been subsumed by other papers (of the same author), or it may turn out that the results were known; or the journals it was submitted to may have considered it "correct-but-not-significant-enough-to-publish", etc. 
The average time between submission of a paper and publication varies wildly. Even assuming that you submit it once, it gets reviewed once and accepted, the time span varies. Time between final acceptance and appearance in print can range from a few months to a couple of years depending on the backlog of the journal (the AMS publishes a yearly list of the backlog of the main journals); time between review and final acceptance depends on the nature of the changes requested and how quickly the authors get to them (anything from a week to several months); time between submissions and initial review can vary as well, though it is considered impolite for a referee to take more than six months for a review. If you end up having to submit a paper to several journals (because the referees deem it "correct-but-not-important-enough", for example) the process can take several years before it finally makes it into print.
My advice would be: check the authors track record. If there is nothing particularly bad about them (a habit of grandiose announcements that are later taken back, lots of errors, etc), then take the paper with double the amount of grains of salt as you take a published paper (I assume you read published papers skeptically as well; if you don't, then you should start), and review the theorems to your satisfaction before "believing them" or quoting them. 
