...I'm addressing the general situation that A) your pet has started destroying your lodger's possessions and B) you want to require the lodger to keep her door shut to prevent a recurrence.
On the one hand, yes, a lodger should make minor accommodations to assist you in dealing with the problem and keeping her possessions safe. However, IMO "you must keep your door shut at all times" is not at all a normal expectation for a lodger. Having property damage as the consequence for forgetting is even more unusual/extreme. If (hypothetically) everyone keeping their doors shut were the rule for your household, it should have been stated up front, before you accepted the lodger...
All well and good except its the OP's bed, mattress, sheets, and duvet not the lodger's. The possessions being ruined belong to the OP. And the OP has already replaced a full set of bedding, twice (which also belongs to OP). The OP is asking for the lodger to take measures to protect the rented items - the bed and bedding are part of the rental.
OK, I was using "lodger's possessions" as shorthand to mean items designated for the lodger's use (e.g., bedding) and that the lodger is expected to take responsibility for if damaged, as well as the lodger's own items that the cat may destroy in the future. I doubt the cat is going to make the distinction between peeing on the OP's possessions designated for lodger use and peeing on the lodger's possessions (clothes, etc.).
However, going back to it being the OP's bed and bedding:
The OP's cat is destroying the OP's possessions--in general, that's the OP's problem and not the lodger's responsibility. The lodger pays to have a room and usable furnishings. The OP's cat is soiling these furnishings--again, the landlord/OP's responsibility to fix. I expect a reasonable lodger to realize that the OP is doing her best to resolve the problem and work with her. However, I also expect a reasonable landlord to realize that she's asking the lodger to change the terms of their agreement to fix the landlord's
problem caused by the landlord's
pet. In general, I think that such effort on the lodger's part deserves some offer of compensation.
If a lodger rents an item and damages it, then it's the lodger's responsibility to fix it. If they rent an item and the landlord damages it, it's the landlord's responsibility to fix it (since they've just deprived the lodger of the item they pay for). In this case, the landlord's pet damaged an item the lodger rents, thus it's the landlord's responsibility to fix.
In the OP's particular
case, I think she's in the clear because she's been accommodating the lodger in other ways. In general
, I think it's her responsibility to fix the problem herself (by finding a way to keep the cat out of the room without relying on the lodger closing the door) or to compensate the lodger for helping her to fix the problem.
As for keeping the door to her room closed? I think that's a perfectly reasonable request.
Unless the request is stated up front before move-in, I disagree. I've never lived in a place where I was required to keep my own bedroom door closed at all times. If my landlord had such a requirement, I'd expect to hear about it up front so I could make an informed decision, especially if forgetting would result in my room being damaged/soiled. If the landlord said up front, "My pet sometimes [chews things, scratches, pees on things, etc.], so you'd have to keep your door shut if you don't want to risk that," then that's a perfectly fine house rule. Springing it on someone after they move in is changing the terms and may require renegotiation of other rental terms.
In fact, actually, it's the *cat* who has made this a requirement--it's simply "how the world works" right now.
The cat's not the one who has a contract with the lodger, so the responsibility still falls on it's owner, i.e., the landlord/OP. The landlord/OP made the decision that "the world" of the household includes both the cat and the lodger, so the landlord/OP has to deal with the problems the cat's behavior causes.
Again, I think the OP's lodger sounds like a real pill and has already taken her own compensation by being so late with rent. But in general, I don't think the lodger is responsible for accommodating the cat's behavior. (And this is coming from someone who adores cats--trust me, I understand that their owners can't just change their behavior at will.)