The answer to question 1 depends on whether or not the couple would need to use the "homestead" exemption to protect their home. To determine this:
How much equity does the home have? (Equity is the difference between the market value of the home and the balance on the mortgage(s)).
If the car is worth more than around $20,000 the Bankruptcy trustee would be able to take it. Therefore, if the car is worth more than that amount, it would make sense to sell it before filing, and then to use the money for everyday expenses.
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When a Bankruptcy is filed, all debts must be included in the paperwork that is submitted to the court. However, the debtor(s) may continue to pay any debts that they want, even though those debts will be discharged, so that there is no longer the responsibility to pay.
Chapter 7: the debtor pays nothing to have debt discharged. This is the preferred Chapter, and would be filed by anyone whose income is below the median for their state, and who does not have substantial property that the Bankruptcy court could take. Total process takes about 6 months.
Chapter 13: debts are discharged after the debtor makes monthly payments to the Bankruptcy court for 3-5 years. This chapter is filed by debtors whose income is above the state median, and/or by those who are trying to hang onto valuable property - such as a house or car with substantial equity.
Chapter 11: filed by corporations (for which a Chapter 13 is not available), and by individuals whose debts amount to more then what a Chapter 13 will allow. Also requires monthly payments.
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