How often do people read the work that they cite? I have the following question:

How likely it is that an author carefully read through a paper cited by him?

Not everyone reads through everything that they have cited. Sometimes, if one wants to use a theorem that is not in a standard textbook, one typically finds another paper which cites the desired result and copies that citation thereby passing the responsibility of ensuring correctness to someone else. This saves a lot of time, but seems to propagate inaccurate citations and poor understanding of the work being cited.
The question is thus about what should authors' citing policy be, and to what extent authors should verify results they are citing rather than using them as black boxes. 
 A: It depends whether I am citing the Poincare Conjecture or the five lemma.  But I agree that either way, one should understand what one is doing well enough to use a result properly.
A: Mostly, I read those part of the papers that I need to cite in my works. Sometimes though, I read them almost completely before starting the paper (to get some ideas and learn the techniques). There are also occasions that I read only a few pages of the paper before citing it. 
A: If it is a recent or not so well known result I am citing then I read completely to make sure it is correct. If it is a well known result that is in essentially any relevant book then it would be enough to know that the result is there and the exposition is good.
A: Often not, and for lots of different reasons.
Citations about related work, background information etc. are often given to give the reader the context of your work, but your work does not necessarily depend on the results therein.
Seminal results from decades ago are sometimes in German, French or Russian, and while I might occasionally struggle through papers in the first two languages if I desperately need to see how something was proved, I will usually take the results on trust, especially if they are heavily cited.
And then, some results are just too complicated, but are sufficiently well accepted that there is really no choice but to rely on them - most group theorists will use the Classification of Finite Simple Groups when necessary, but few, if any, will understand the full details of the entire proof.
On similar lines, I cannot possibly read the Robertson-Seymour series of papers on Graph Minors. Yet I believe the main results in them, and if and when I refer to the fact that any minor-closed class of graphs has a finite set of excluded minors, then it is pretty much obligatory to refer to the relevant paper in the series. 
If correctness is seriously in doubt, then it would presumably be necessary to work back not just to the papers you cite, but to the papers they cite, and so on.
A: I normally do. Right now I'm facing a tough choice though: to read David-Semmes book in honest or to write something like "We prove that A implies B. The reader can juxtapose that with the claim on page ... in [] that B implies C" instead of "We prove that A implies C" in the introduction. Being as lazy as I am, I am inclined to go for the second option but that will certainly make the paper less "sexy", so my co-authors do not feel very happy about it.
However this shows how you can avoid both reading the papers you refer to and the uncertainty about whether what you declare proved is actually proved: separate the part you prove from the part you refer to in a crystal clear way and take credit for the reduction only rather than for the full statement (which, frankly speaking, is as much credit as you can really claim anyway).
The correction mechanism Nik mentioned works primarily in the way that most things just go unnoticed because nobody reads those papers or uses them in any way. When something is really important, it gets a lot of attention and somebody finally straightens things out. However, that doesn't happen fast and I have learned it hard way. My 2002 Duke paper joint with Treil and Volberg on the system Tb theorem has an error in the proof. It had been cited a lot of times before the error was finally spotted and corrected by Tuomas Hytonen around 2010. This also shows that an erratic argument isn't always useless or fatally flawed. Sometimes it is just an "incomplete proof". To my shame, I should also mention that it was one of the cases when I didn't read the final draft carefully and relied on my co-authors to do that. Apparently, they had a similar attitude...
A: We are more inclined to read the details when we are young, for several reasons. First, because we are more energetic than when getting old. Second, because we are a little bit idealistic, while everything tends to become relative after we have seen all kinds of behaviors in science. Last, because we are often single-minded at the beginning of our career; we do not have too much to read and we are not disturbed by administrative duties.
Of course, It is of great importance to have a clear and complete understanding of the papers that we cite. This is why young mathematicians play such an important role.
A: This may vary depending on the field you work in and what kind of papers you're writing, in addition to your personal style as a mathematician. I mean whether you bother to check the details of a citation probably has a lot to do with how much effort it costs you. In my line of work the citations are usually to other papers within my area of expertise and I do in fact try to look pretty closely at what was done and whether it really gives what I need. But there have been times when I needed something outside my comfort zone and was content to take the word of an expert that so-and-so's theorem did the trick.
On the point about propagation of errors, it seems to me that there is a sort of correction mechanism. If a result is being cited incorrectly, eventually it may lead to a contradiction which would then be unraveled by tracing the problem back. Does anyone have any examples of something like this actually happening?
Oh, I should add that some portion of my citations are "courtesy" citations along the lines of "so-and-so did something related". I'm afraid to say in cases like these my reading of the cited paper will sometimes have been very superficial.
A: [offtopic] Since I cannot comment, let me just throw in an old story I heard from my professor. Some time back, a paper by Einstein and Preuss was being cited all around. Now, we all know some names that collaborated with Einsten, but this Preuss is kinda unknown. Turns out that the journal name of the reference
Einstein, A. (1931). Sitzungsber. Preuss. Akad. Wiss. ...
after some citations, got to be promoted to coauthor. NICE!
Here's some (german) reference: http://de.wikipedia.org/wiki/S._B._Preuss
Now just to account to the statistics, I know some people that skip the reading of some papers to present seminars and talks, but I think they do check the stuff before writing something up. As to me, I try to read some stuff and then check the references and references of references until I give up. But that doesn't matter, since I'm far from publish anything at all, as it seems. Cheers.
A: I like papers with big reference lists. Sometimes these citations lists are even more useful for me than papers themselves, since I can find some others papers. And this describes context and rises new questions for further work. 
So since I like big reference lists in papers by others, I also try to include many references, 95% of them just "someone have done something related", and of course I do not read carefully such papers, just may be understanding what question has been asked and what is the context of this question.
However there are some (I remember 2) cases where I need to heavily rely on the results by others without deep checking the proofs (since it would cost too much effort).
I do not like doing so, but my experience says me that it should sometimes be done.
In general I think it much depends on the area you work, in my field it seems to me simple useful constructions are valuable, rather than complicated proofs.
I like what Igor Pak wrote here : Presenting work in progress 
let me quote: 
"For example, in Enumerative Combinatorics and Discrete Probability, two areas close to me, these priorities are sort of opposite. In the former, there are very few open problems. A nice new formula or a new bijection construction, even if only conjectured and checked by a computer, is already a lot of progress. Once you convince yourself that you can finish the proof, you can start giving talks - people will trust your judgement.
However, in Discrete Probability, there are lots of open problems and conjectures, often delicate and technically difficult. I would advise NOT to speak about your results until the proofs are fully written and carefully checked by somebody. This might work once or twice, but eventually there will be a seemingly trivial mistake which you overlooked in the first draft. Unfortunately, often enough such mistakes can completely destroy your proof."
So I think in the first type of areas deep checking is not relevant, but in the second it is very necessary. (In my (subsub)field situation is like 1, which makes my life more easy).
A: I think it was Thom who said was that it was immoral for a mathematician to base his work on results he/she did not understand. Sometimes I think this attitude would kill off whole areas of research, but it is probably the way to go. In my case it makes a difference if I need someone's work for to prove a neat corollary or as an argument in the proof of a main result. 
