Only if they had mortgage insurance.
Actually, the home owner pays the home owner's insurance. The lender has an escrow account. This is in additional to the payment of interest and repayment of principal. The escrow account pays the taxes and insurance. The escrow account pays the taxes so the government does not seize the property. The homeowners insurance pays in case the house burns down. So, you pay into the escrow account, and if your house burns down, the lender gets the insurance money. You would not pay a mortgage on a burned down house and the bank knows that, so they have you pay into the escrow account and they pay for the insurance.
There are two general types of policies, or combinations: lender's insurance (which pays the lender to cover its loss in security interest) and owner's insurance (which pays the owner in case of defective title).
No, homeowners insurance pays for damages and losses due to certain hazards listed on the policy. Typically, Fire, wind, Hail, falling objects etc, But it will not pay the mortgage note nor pay the house off due to the death of a buyer.
It would be possible to write an insurance policy that way if you wanted to, however, normally a life insurance policy pays a fixed amount of money (known as the death benefit) to a chosen beneficiary. If the beneficiary then wished to use that money to pay for a home, that could be done.
The rule of thumb is that the owner's insurance pays first and, if that coverage is inadequate, the driver's car pays.
Usually the insurance policy of the owner of the car is primary and then if the driver of the car has a policy of their own then it is secondary.
Credit Life Insurance.
In a condominium usually the board pays for property insurance on all common areas. An individual owner pays for 4 walls, the ceiling, the floor, the inside of the door, and everything in between.
It really just depends on your lease. In some areas it is customary for the owner to provide the structures insurance while in other areas it is commonly part of a lease agreement that the tenant will provide it.
renter's insurance
The question does not really involve "should". The direct answer is "no". Using life insurance as an example, the owner of the policy is often the person who pays the premium. The insurance contract gives the owner various rights, such as to initially designate the beneficiary, change the beneficiary, pledge the policy as security for a loan, and other acts. The insured is the person whose life is, well, insured. Stated otherwise, this means that when the insured dies, the insurance company generally pays the death benefit to the beneficiary.
Different insurance companies may have different insurance policies, so contact your insurance provider and ask them if their insurance pays to replace a cat.