Are corporations separate between DUST 514 and Eve, or can a merc join an existing Eve corp? If separate, can a DUST corp have the same name as an Eve corp? For instance, could I train corp mgmt on my merc and open a Du'uma Fiisi Integrated Astrometrics in DUST despite it existing in Eve?
Corporations are seamlessly integrated in terms of membership. If a Dust merc joins a corporation he is joining an EVE corporation, essentially, even if there are no EVE players in it. The new NPC corps for Dust players are an exception.
All of us share the same server, and corps, alliances and player names for both games are in the same namespace in the database. If an EVE player takes the name "Samuel Poopyhead-Fartface" for their character, that name becomes unavailable as-is for use as a corporation name, alliance name or Dust character name. You might find it with different punctuation or spelling, but not as it is, until the character in question is biomassed and the name freed up.