Setting up a website that has the potential to be abused and used for illegal services. How do sites like craigslist or facebook avoid getting in trouble for such potential? Could my site be liable for these issues?
First off, you have not told us what your site does so we have no way of even speculating the legality of it. Having potential to be used illegally can create a liability, you are required to take reasonable efforts to circumvent them. You are generally only liable if you were negligent. For instance, if you run a chat room, child predators can hang out there to go after kids. If this just one day happens and a child is kidnapped, you are not necessarily liable. However, if your site had a history of this sort of behavior that you did nothing to try to stop it, you may be found partially liable. What type of site are you running? Some sites have more risk than others. For instance, The Pirate Bay could not run in America even though it can be used to get legal torrents (such as Linux distributions).
In theory, yes...but always depends on the situation. It would help to protect yourself with customer verification, monitoring, auditing etc. and a good 'conditions of use' document. I assume you are talking about some sort of hosting?
It's called the DMCA - the law provides web site owners the ability to have immunity from prosecution if they comply with the DMCA, essentially taking down illegal and infringing content in a timely manner after being notified.
i agree with xanth, you must stick with DMCA policy, in order to keep yourself safe from legal issues. You have every right to delete illegal contents and to make a Terms of Service(TOS) page for your users or customers to see what is allowed and what is not.