First, everything is negotiable. Per law, your lease may no longer be valid if the mortgage was in place BEFORE you signed your rental agreement. More to that on the website below.

In regards to "cash for keys" the amount you can get paid for voluntarily moving out is definitely negotiable, too. Let's say you find another place to live and they are asking for a $1,000 deposit and you need a moving truck that's going to cost you $100. Get it in writing and present your case to the PAS rep - more likely than not they will give you the $1,100 in return for you leaving the house in an orderly manner and leaving no trash behind etc.

It is cash for keys. You can however negotiate more time. Also under the new Obama law, if you have a lease you can stay in the property for the duration of the lease/

They are probably offering you "cash for keys" which is something the foreclosing bank does to get you out of their property without having to file a formal eviction on you. Basically, it's cheaper to offer you money to leave and to exit with the house in good condition than to evict you and risk having a tenant leave the home in need of a lot of cleanup.

I believe your given this option so that they will not have to file an unlawful detainer. Signing may not mean you have 10 days to move. Sounds like they just want to know you will not give them a problem when the time does come for you to vacate. Check the link I posted for more clarification.