I know of two exceptions to the above - the Wall Street Journal and USA Today. As far as I can tell both are printed in many cities for prompt delivery nation wide. The Wall Street Journal only covers financial news. USA today tries to be a national newspaper, so it doesn't cover any area of the US very well.
The New York Times is also delivered nationally. It costs an arm and a leg, but you can get it. It's also sold in just about every Starbucks I've ever seen, and in bookstores.
I am not sure about the range of delivery, but the Los Angeles Times and the Washington Post are also major newspapers with national readership. I think the Chicago Tribune may be pretty widely read as well.
The Wall Street Journal may cover financial news primarily, but it is not just financial news. They certainly cover politics a great deal of the time, and have other non-financial news stories. A lot of people read the WSJ as their primary source of news.
From my own personal experience and observations, most major metropolitan areas in the US have at least one newspaper, and some have more than one. Most places that can properly be called "cities" (rather than towns) also have their own newspaper, most of which are dailies.
However, a lot of the content in smaller newspapers especially, but even in large newspapers, comes from places like the Associated Press or Reuters. Typically, news that is international or national will not come from local reporters. Most of the columnists in the Op-Ed pages will be nationally syndicated as well, although there may be some local contributors every so often. Only newspapers as large as the New York Times or Washington Post actually employ Op-Ed columnists directly, and those people are usually the ones whose columns show up in other newspapers.
Since so much of the content in most newspapers isn't actually produced by the newspaper itself, they tend not to "lean" especially obviously in one direction or another. There are exceptions to that (which I won't name here, to avoid contention), but most newspapers' editorial boards have political leanings that become obvious during election season. Editorial boards typically endorse specific candidates running for office, and who they endorse can tell you a lot about what their political views are.