If a new NPC from a mod has a Sleep AI Package that referers to a bed owned by a vanilla owner will the new NPC sleep in it?
The bed's owner's (innkeeper) sleep package points to a bedroll, and the bed is not referenced by anything except the new NPC's sleep package.
The new NPC is not assigned to any faction.

If the AI Package fails due to ownership, will it work if they are in the same faction. Or does the bed in that case need to be owned by the faction?

Pretty sure that the bed also needs to be owned by the faction, this is why the Count of Anvil spends his nights standing next to his wifes bed.

...which is fixed in version 3.3 of the UOP. There were a couple of these cases that were fixed in 3.3, so PetrusO you might want to check that out to see how it was fixed. I think in general the ownership of the bed was changed to a more generic faction so that all the NPC's of the faction had the option to sleep in it. You might have to tweak the mod in question so that the NPC is a member of the same faction as the bed ownership as well though...

I see the bed is now open for everyone in the AnvilCastleFaction.
But looking at the Count of Anvil spoilt me -

If you're looking at the Count of Anvil, it should be "was", not "is".

I've always done shared beds by making a specific new faction for the bed and using a SetFactionRank to assign the occupants. If it's an existing bed, use SetOwnership too, and you don't need to edit any of the original forms. Works for simultaneous use of doubles, or alternate use of singles.