In the real Navy, you can command a vessel at the rank of Lt. Commander. It'll be a smaller, likely older vessel but it's still a command. It's the higher class of ship that require higher-ranks (Aircraft Carriers, for example).
Correct - and, IIRC, it is nautical tradition that whoever is in command of a vessel is referred to on the deck as "Captain", irrespective of rank (at least, for anyone who holds a grade below an actual Captain). This was touched on in a very cursory fashion when all the bridge officers were addressing Saavik as "Captain" in the Kobayashi Maru simulator.