A precinct is different than a polling place, but are used interchangeably; typically, there are several - even a lot - of precincts per polling place. Oftentimes, the media switches the two.
Not in South Carolina. The rule here is that each precinct has its own polling place. Also, no precinct is supposed to have more than 1500 registered voters, so as to keep from having long lines or parking problems. That last rule sometimes gets honored in the breech, but the State did a rather complete job of splitting oversized precincts up in time for the 2004 election.
I think several Southern states do this. Some break larger polling places into alphabetical last names, for some reason. Florida
does differentiate, though, at least in larger counties with denser populations - where it makes no sense.