Example: I have a document containing 1,000 abstracts of research articles from different publishers. Am I allowed to distribute this document for free online? If not, has anyone being sued for doing so?
Am I allowed to distribute this document for free online?
I'm not a lawyer, but I doubt it. Copyright certainly applies, but you might be able to argue that this falls under fair use, depending on what you are doing with the abstracts. As I understand it, some things are definitively covered by fair use, while others may fall in a grey area. If this matters to you, then you should consult a lawyer, because no one else can give you a definitive answer.
If not, has anyone being sued for doing so?
In the U.S., I doubt you'd have to worry about being sued or facing criminal charges for this sort of copyright violation in practice (although I repeat that I am not a lawyer). The most likely scenario is a DMCA takedown notice. If your university or ISP receives such a notice from a publisher, they will remove the file from your web site. You could try to convince your university to take a stand based on fair use, but it's not at all clear that you would succeed. Even aside from whether it really is fair use, universities can be risk averse and may well decide it's not worth the effort to contest the issue.
Perhaps you could restrict your attention to papers with a suitable Creative Commons License. For example, PLOS journals use the CC BY license, which allows this sort of reuse (with attribution). If you just need some collection of abstracts, for example as a test case for text processing or data mining algorithms, then this might work fine.
On the other hand, you may want to use very specific abstracts. For example, you might want to compile the abstracts of all the papers you can find in your subfield. This seems riskier, since big commercial publishers typically don't want competition for their search tools. You could always try asking the publisher for permission (I have no idea how likely they would be to grant it).