If he is a Medicaid applicant or recipient, there is a "Five-Year Look-Back Window" so that Medicaid can be repaid for the expenses it covers for him. Hopefully the Power of Attorney you have is a 'Durable' Power of Attorney so that it is still effective when he is incapacitated. I'd recommend you sit down for a consultation with a California Elder Law attorney - preferably one who is a member of NAELA. Seek out counsel using Avvo.Com under Find-A-Lawyer. Good Luck in this case.
I am sorry that you and your family are facing such struggles.
It is probably better to address this in two stages - first the security of your father's retirement funds and then the obligations on his home and car..
You have indicated that your father's retirement funds are needed to pay for the senior care facility. Retirement funds in assorted accounts are generally exempt from execution by creditors. Mere savings accounts are not. You have knowledge of how your father's retirement accounts are held and this is a consideration in dealing with creditors.
If your family decides that it would like to sell your father's home you should secure a knowledgeable real estate broker who can assess the market and advise you on the sale. If the home is over-encumbered a short sale can be used that would require a waiver of any deficiency against your father by the lenders. You could also consider renting the home as a way to pay the mortgage (obviously depends on whether the rent could pay the mortgage and associated expenses of home ownership (taxes, maintenance and insurance).
The car is a different issue. You will need to assess whether the car has equity and whether it can be sold for enough funds so that the lien against it is paid off.
Lawyers familiar with estate planning issues should be able to guide you in all of this.
This is a general answer only and you should seek the advice of counsel to address facts specific to your circumstances.
Because you have the durable power of attorney for your father you should examine his financial situation in its entirety and decide what is best for your father given that he is now in a senior care facility. Starting with the home: If you do not anticipate that he will move back into his home then you should decide whether to rent it out or sell it rather than leave it vacant. Other possessions including the car need to be managed so as not to result in uneccessary financial loss. The issues of what happens after your father passes depends on his estate planning and are not important right now. I suggest you contact a real estate broker to see what sale of the house could mean and a property management company to see about rental value and look at the blue book value of the car versus the amount owed as starting points. Consulting with an attorney to help you manage your father's needs even in a limited capacity is a good idea for both you and your father if you believe you cannot manage things on your own. Good luck.
Since your father has terminal cancer and is in the nursing home you should consider hospice care. Taking the necessary steps to qualify him for Medical while preserving as many assets as practical is an option, but timing is critical. You need to see an experienced elder law attorney who knows the ropes in Medicaid/Medical planning. I suggest you search AVVO and the National Academy of Elder Law Attorneys at www.naela.org. You need good professional help and you need it now.