Americans are guaranteed freedom of expression and obscenity laws are something that attempt to curtail this expression. Censorship is explicitly anti-liberty, and actions and words that may offend others should still count as protected speech, as indeed ALL speech should be protected under law. Just because something seems obscene doesn't make it invalid.
Yes, the Bill of Rights prohibits censorship laws, because it is clearly stated in the First Amendment. The First Amendment states that the right of the people to free speech and freedom of the press shall not be abridged. That means free speech in all of its forms. And the press is not any different just because it is now online.
I do not believe the Bill of Rights prohibits censorship laws. The United States has used censorship in many ways over the years, but it's always been thought to be minimal. Censorship is something Americans don't have to deal with as much as some other countries, but it is thought that it may be an increasing problem.
The Bill of Rights does not prohibit censorship laws. What the Bill of Rights does is make sure censorship laws won't overtake the basic rights of people. There are still times when censorship is not only allowed by the Bill of Rights but would be encouraged by the Bill of Rights.
The Bill of Rights does not prohibit censorship laws. The Constitution protects certain civil rights, but it does not prohibit lower federal or state laws from making reasonable limitations on the Bill of Rights. In practice, there are several legitimate reasons to censor the media, for example, libel and child pornography.