We do have a rule that if "the" is part of the title, it should be capitalized. Some follow this rule, and some do not. Often, it depends on guidelines that are almost bizarre. The best rule is to check with the newspaper itself, and see how they want it. "The" is often dropped from the title because many sentences are confusing when this isn't done. If, for example, you write, The New York Times editor. . ., it sounds like there's only one editor. This means you write New York Times editor. . ., and it's correct.
As for italics, this is even worse than the guidelines for "the". Style guides usually say to italicize them, but I'd say far more don't follow this rule now than do. Even the newspapers themselves very seldom italicize the name of a newspaper. You can probably find a thousand articles where there are no italics for every article that does use them.
I stopped using italics for newspaper titles years ago, and every editor lets this stand because they do the same.
