Whether or not you can both be on the tenancy agreement will be up to the landlord/rental agent. I don't see why she could not be.
Yes, you would need the letter giving permission for her to live there.
As I previously said if you are going to submit soon, you could use the current address (if landlord gives the permission letter)
I am not a regulated immigration advisor and am only providing my own personal views and opinions. This should not be relied upon as advice.