By most accounts San Marino is the the old surviving country. It was founded on September 3, 301 AC as a republic. Despite being founded in 301 AC, the constitution was not written until the year 1600, and in 1631 it was recognized as an independent country by the Pope.
Sorry, we don't allow trivia-hunting questions.
I believe Iceland has one of the oldest systems; dating from ~1000AD if memory serves? I haven't got my sources to hand I'm afraid; please correct me if I'm wrong on that!