Can newspapers print whatever they want and not get sued?

Nope.

There are 9 things a newspaper cannot publish, they are called the unprotected nine.

obscenity,

fighting Words,

defamation,

copyright infringement,

invasion of privacy,

encouraging others to break the law,

clear and present danger,

advertisements for illegal products,

inciting a riot,

If you wanted to sue a newspaper it would most likely be for defamation, or Libel. That can be hard to prove because you have to show that the newspaper intended to harm you (malice), and that your reputation was damaged because of the story that was published.

Hope that helps :D