The nursing home can bill the deceased's estate. If the deceased had any assets when they died, the nursing home can collect unpaid bills from the remaining assets.
It depends. The family members should speak to an attorney, or the attorney handling the estate. Generally, one is not liable for the debts of another unless it is subject to a written agreement, however, a spouse may be liable for "debts incurred on account of necessaries furnished to either spouse or to a member of their family."
You should speak to an attorney to review the specific facts related to this matter.
This answer is provided for general, informational purposes to the "hypothetical" questions posted and does not create an attorney/client relationship. It is recommended that you speak to an attorney to discuss your specific legal issues.