There actually are lots of safe harbor protections for common carriers. Lack of editorial oversight is why sites like YouTube can qualify for these protections, but sites like Gawker wouldn't be able to. (Even though they ultimately got shut down for ignoring a court order or whatever.)
Sure, but that's within safe harbor rules. If they went and moderated content and comments in a non-reactive way, say by banning commenters whose opinions they found incorrect, that would be legal but could jeopardize their safe harbor status.
And as it states there, they also have censorship based on their terms of service, where they prohibit the posting of videos which violate copyrights or depict pornography, illegal acts, gratuitous violence, or hate speech.
