My husband and I are on the same SE account, so we didn't think to ask that question. We just chose which character we wanted to enter and took the screenshots in her Mog House since only one character per SE account can enter.
The theme is "Spending Time with a Special Someone" so I would think it would be okay as long as the /names aren't showing, but it would be nice to get an official word on this one. But seriously, it's kinda hard to spend time with someone if the person isn't there. That might be why they specified that the first screenshot clearly shows your character's name and mog house.
I have no idea on this one. If it was me, I would go by my SE account since the contest restricts you to one character per SE account.