If the check they give you says "paid in full", most courts accept this as the end of the discussion. Your endorsing the check means you agree to this as final payment. (I'm not a lawyer, but I love watching "The People's Court").
Not really, the memo does not end their dept obligation. Its not like you could make a car payment and write "paid in full" on the check, then once the check clears declare your debt obligation to the bank settled.
If they want something that says ?paid in full? from you, don?t give it. What would need to happen is you present them with a full invoice when make an offer to settle you tell them that you will still show the balance on then invoice so that you can write off the loss on your taxes. If they baulk, play it up and ask why they would begrudge you the loss for tax purposes. Don?t tell them that you intend to file a 1009 on them later.