1) As a CFI you MUST hold a valid TSA certificate in addition to your
CFI certificate. Check out AOPA's page for info on obtaining this
certificate. You are technically illegal in giving flight instruction
without a TSA certificate.
2) There is no requirement that you prove citizenship for a BFR. Check
out http://www.aopa.org/tsa_rule/ . TSA only applies to new ratings.
3) There is no requirement to keep any paperwork (other than the CFI
keeping his TSA certificate). As a CFI you simply write an endorsement
in the student's log book and forget about it

I certify that [insert student's name] has presented me a [insert type
of document presented, such as a U.S. birth certificate or U.S.
passport, and the relevant control or sequential number on the
document, if any] establishing that [he or she] is a U.S. citizen or
national in accordance with 49 CFR 1552.3(h). [Insert date and
instructor's signature and CFI number.]

Signing this endorsement is a SUBSTITUTE for record keeping.

-Robert, CFI (and TSA certified!)