The Dragoon storyline goes into this in detail. Suffice to say: it's not exactly being used for its intended purpose, and Nidhogg is not too chuffed about idiot hominids having it.
Anyway, yeah, all of the currently-extant city-states are about 600-700 years old, with the exception of Ishgard being a little bit older. Gridania and Ul'dah evolved out of somewhat older civilizations (with some growing pains), and Limsa sprang up whole-cloth after a major shipwreck stranded a bunch of Sea Wolf Roegadyn on Vylbrand.
Not quite. Limsa's been around for a while, but as a town founded by pirates and led by a large number of different administrations over the years, it's been many things over its existence. Generally, it's been seen as a haven for pirates and outlaws from the rest of Eorzea, although how true that's been has varied over the centuries. Merlwyb's just been going much, much further than previous admirals in properly "washing Limsa's face" and turning it into a proper, respectable state. The Rogue questline actually goes into a fair bit of detail about the tensions this is causing, and the questline for unlocking and doing Hullbreaker Isle also elaborates on why Certain Influential Figures are actually cooperating and do want to see Limsa's old, bloody course changed a bit.