Is it legal to fly in RVSM airspace with an inoperative autopilot?
About whether or not you can fly with an inoperative autopilot, take a look at §91.180:


  §91.180   Operations within airspace designated as Reduced Vertical Separation Minimum airspace.
  
  (a) Except as provided in paragraph (b) of this section, no person may
  operate a civil aircraft in airspace designated as Reduced Vertical
  Separation Minimum (RVSM) airspace unless:
  
  
  The operator and the operator's aircraft comply with the minimum
  standards of appendix G of this part; and
  The operator is authorized by the Administrator or the country of
  registry to conduct such operations.
  
  
  (b) The Administrator may authorize a deviation from the requirements
  of this section.


You also can't say that your aircraft is allowed to operate in RVSM airspace:


  Section 4. RVSM Operations
  
  (a) Each person requesting a clearance to operate within RVSM airspace
  shall correctly annotate the flight plan filed with air traffic
  control with the status of the operator and aircraft with regard to
  RVSM approval. Each operator shall verify RVSM applicability for the
  flight planned route through the appropriate flight planning
  information sources.
  
  (b) No person may show, on the flight plan filed with air traffic
  control, an operator or aircraft as approved for RVSM operations, or
  operate on a route or in an area where RVSM approval is required,
  unless:
  
  
  The operator is authorized by the Administrator to perform such
  operations; and
  The aircraft has been approved and complies with the requirements
  of Section 2 of this appendix.
  


If it's not operational, the aircraft no longer qualities for RVSM, and therefore must be provided with 2000 feet vertical separation (each way) between other aircraft. Alternatively, ATC can refuse entry into RVSM airspace.

Regarding having the autopilot engaged, Appendix 4 of AC 91-85, Section "5. In-flight Procedures" states: 


  e. An automatic altitude-control system should be operative and engaged during level cruise, except when circumstances such as the need to retrim the aircraft or turbulence require disengagement. In any event, adherence to cruise altitude should be done by reference to one of the two primary altimeters;