If you're not married, then I think technically you can, though you'd need to convince them that you were no longer living as a couple. That means properly living like housemates - separate rooms - ideally with locks on the doors, eating separately, and separated finances. I think it's possible, but possibly difficult if you're short of space.
Yes you can. My first claim included a short time when we were in the same house. I spoke to them about it and they said this was ok as we had technically separated and had started divorce proceedings. (I was still married at the time.)