As far as an FFL keeping the NICS paper form, that is an invitation for someone other than the FFL to get the information and use it against you. A SS# combined with the other information on the form such as date of birth, address, etc. is more information that I want to be stored by a third party and which no assurance can be given that this will always be within the control of the FFL.
Since it is not likely two persons with the same name were born on the same day and live at the same address, the Feds at the NICS data shop should be able to distinguish you Vs. a bad guy and this is another reason not to give out your SS #.
My advice: don't do this with a NICS application or for other "voluntary" requests; I beleive it is specifically discussed in the federal law and only your employer and the IRS can ask for this. Banks and brokers are given this info voluntarily to avoid witholding 20%for taxes on interest and dividends