Do we have any details on whether both accounts that participate in Co-Op play will have to pay for the Necro DLC? If not, what happens when the account without the license starts up Diablo3 for some solo play? Will their Necromancer become unavailable?
You only need to buy it once, multiple accounts on the console can use it.
This is correct. Also bear in mind that users can only change their Home/Primary console a couple of times a year, so there's no need to take this action if you're, for example, visiting a friend's house to play the Necromancer with them.