Will courts consider a cashed check as a written agreement for the payment of damages to a car in a car accident?

Write on the memo line on the front of the check and on the back of the check up where she has to endorse it "full and final payment arising out of 6/18/09 car accident." If she cashed the check, it's like a release, she will have accepted your offer as settlement and can't then ask for more money. You can always pay her more, but won't have to if she cashes this check.

A check isn't a written agreement, but is evidence of a oral agreement, as are emails. You could even do a written contract by email if you send terms and she accepts.

As for your suspended license and her reporting the accident, you shouldn't have been driving, and you can't ask her to not to something she's legally supposed to do, and if the accident damages exceeded the amount for which she was (or you both were) supposed to report the accident, well, you both should obey the law.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.