Under what conditions can a non-RVSM aircraft or pilot fly in RVSM airspace?
According to the AIM Section 4-6-8, the language to use with ATC is simply, "Negative RVSM."  You have to say it on the request for an RVSM flight level, flight level change, or climb/descent through RVSM airspace.  You must also say "Negative RVSM" on check-in if currently at an RVSM flight level.

NBAA provides a guide for navigating the RVSM approval process.  This guidance is fairly new, and indicates that approval through a prior operator of the airplane is acceptable.  Your FSDO will make the authorization.

See also FAA RVSM Documentation