Are you talking historically or in the Shakespeare play? I don't know the historic answer, but in the play, it's a bit more complicated. He stabs himself after the battle but doesn't die instantly: he's carried to Cleopatra's hiding place to die there.
A very quick Google search on "Actium" (without inverted commas) produced this site, http://en.wikipedia.org/wiki/Battle_of_Actium whose historical notes say that Antony and Cleopatra escaped from the battle, and both committed suicide a year later.
Yes. Plutarch says that 13 years after the murder of Caesar, Antony and Cleopatra were dead by their own hands. The location was a palace believed to be on the present day waterfront of Alexandria.