Crews are usually just named for the captain's epithet, so unless some exception was made for us we'd be known as the Redcoats. Of course, that does make us sound like colonial British soldiers. :P

As for ship name, since it has a dragon figurehead we can be lazy and call it the Going Ryu, or go ironic with Going Tora (tiger). Or get more metaphorical or poetic with it, like the Sky King (which would be incredibly humorous if/when we find a sky island).