I'd answer "maybe" to all questions. It's also worth noting that the company airship has two functions. The Diadem is one. The other is just where you send the ship out on its own to collect stuff (much like a retainer venture). I haven't done the math on how long it's likely to take for that to actually be profitable, though.

As for the Diadem: Firstly, you can just get there via Ishgard without needing a free company airship. Secondly, it's worth knowing that you'll likely encounter others in the Diadem regardless of who you go with. It's not a private instance like a dungeon.