No—but there was a family connection, as "Lanza" was his mother's birth name. He was born Alfred Arnold Cocozza, the son of Italian immigrants Antonio Cocozza (1894-1975) and Maria Cocozza (née Lanza) (1902-1970). While still a high school student, Lanza was already experimenting with alternative names for a stage career, as pages from his school notebook reveal. He finally settled on his mother's birth (or "maiden") name, together with the masculine form of her first name. Called "Freddie" by his parents and childhood friends, the tenor was known professionally as Mario Lanza from 1942 onwards, and on October 7, 1948—exactly eleven years before his death—he legally adopted that name.

Five feet, seven and a half inches, or 171.45cm. This is according to his December 1942 army induction records, which can be viewed here. There is also ample photographic evidence that Lanza was not a tall man, although this has not prevented some of his more fervent admirers from insisting that he was really 5' 10½," or almost 180cm.

Yes to both questions. Lanza married Betty Hicks (1922-1960), the sister of one of his Army friends, on April 13, 1945, and the couple remained together until his death. They had four children, three of whom died prematurely: Colleen (1948-1997), Ellisa (1950- ), Damon (1952-2008), and Marc (1954-1991). Betty Lanza outlived her husband by just five months, dying of apparent asphyxiation following a prolonged period of depression.

Answer

In South Philadelphia on January 31, 1921. However, Lanza's birthplace was often misreported during his lifetime as New York City, with a variety of incorrect birth dates also provided (most often, 1922—a date that was probably encouraged by MGM's publicity department). Lanza himself variously claimed to reporters to have been born in 1922, 1923, 1924, and 1925. In an interview with Howard Thompson of the New York Times on May 6, 1951, the then 30-year-old Lanza cheerfully alluded to his habitual fibbing, stating that he was twenty-nine—"And twenty-nine I'll remain for at least another ten years." In this respect, he was carrying on a family tradition, since his mother shaved almost three years off her own age, always insisting that she had been sixteen when he was born. (She was actually in her nineteenth year.)

No, he was an only child. He later stated that because of the loneliness he sometimes felt while growing up, he often dreamed of having a large family of his own.  

Lanza died at the Valle Giulia Clinic in Rome on October 7, 1959. No autopsy was performed, but his death was almost certainly due to either a pulmonary embolism—arguably the most likely scenario due to the phlebitis he was suffering from at the time—or a massive heart attack. A third (though less likely) possibility was a cerebral hemorrhage. See Mario Lanza: A Fatal Zest for Living, by Armando Cesari and Philip A. Mackowiak, for more information. 

The absurd rumor that Lanza died at the hands of the Mafia is addressed here.