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How many children did he have?: Alexander and Elizabeth Hamilton had eight children. Remarkably for the time, all but the eldest, Philip, lived well into adulthood. Philip died in a tragic duel in 1801, defending his father's honor. Alexander only gave his son one bit of advice for the duel, that being to fire into the air.
As Hamilton had a number of children, yes there are many descendants of Alexander Hamilton and Elizabeth Schuyler. Not all of them carry the surname of Hamilton as he and his descendants had daughters who married and had children. As Pat Sanftner stated, Douglas Hamilton is the first born son of the first born son (etc) and the 5th Great Grandson of Alexander & Elizabeth Hamilton.