Could we sell the house for $170,000, settle with the 2nd mortgage company for $10,000, and use the rest as a down payment on a new home?

You can settler the second lien for whatever the lender is willing to accept in exchange for releasing their lien against the property. I don't really see why they would want to reduce the amount by 80% if there is value in the property. They still retain their lien and could choose to foreclose if the wish to.

You can do as you like as long as the lender agrees to accept a lower amount.

If you can work out an arrangement with the 2nd mortgage company you're able to negotiate any type of deal you want, your prior bankruptcy has no affect on that. The reaffirmation agreements only mean that you are liable on the loans if the home was to be foreclosed. Even though you did not sign a reaffirmation agreement, you still owe the full $50,000.00 on the 2nd mortgage but they cannot try and collect any deficiency against you if the home went to foreclosure. Typically a lender will not negotiate a short sale-where they agree for you to sell the house for less than what is owed on the mortgages-and allow you to keep a large portion of the proceeds. Because you did not sign a reaffirmation agreement on the 2nd mortgage, If you let the house go into foreclosure you will never owe the 2nd mortgage another penny. Depending on which state you live, the same could be true of the first mortgage as well.