In its very first issue, the paper said, "We publish today the first issue of the New-York Daily Times, and we intend to issue it every morning (Sundays excepted) for an indefinite number of years to come." It is one of the few papers still around from that period. The Times did in fact begin publishing a Sunday edition, during the Civil War.
Most New York City newspapers in the 19th century had their offices on Park Row, right near City Hall, causing people to refer to the street as Newspaper Row. The Times moved to Longacre Square at 42nd Street and Broadway in 1904, and convinced the city to rename the area Times Square, after the paper. (The New York Herald had done something similar a few years before with its move to Herald Square.) The Times is currently housed at 229 West 43rd Street, but plans to move to a skyscraper currently being constructed at 41st Street and 8th Avenue.
Reference: Wikipedia: The New York Times