What information am I legally required to provide during a FAA ramp check?
AOPA has a good article on this topic:


You're required to present pilot and medical certificates (plus logbook if required for flight)
The inspector may not detain you
The inspector may not board the aircraft without informing you


But an inspector is authorized to check the following items:


The airworthiness certificate
The aircraft registration
The operating handbook
The weight and balance information
The minimum equipment list (if applicable)
Aeronautical charts (if applicable)
The general airworthiness of the aircraft
The ELT battery
A VOR check
The seats/safety belts.


In general, saying as little as possible seems to be a good plan:


  If the ramp check is due to a possible violation, anything you say or
  do may be used against you.
  
  If you have enrolled in AOPA's Legal Services Plan, call AOPA's Pilot
  Information Center at 800/USA-AOPA immediately. The consequences for
  even minor infractions can be far more serious than you might think.


As for a search warrant, a ramp check is an FAA administrative process, not a law enforcement action. The issue of law enforcement checking private aircraft is a hot topic currently and AOPA provides a checklist in case you are stopped for search. Their recommendation is to refuse the search (unless they have a warrant, of course) but not interfere if they go ahead and do it anyway.