Are there any rules requiring hands/feet on the controls at all times?
There is nothing I can find in the FARs requiring that a pilot keep their hands and feet on the controls at all times.
Such a requirement would be impractical and dangerous in many situations (such as when writing down a clearance in the air: For most people if you keep your hands on the controls while writing you will have a tendency to push, pull, or twist them, resulting in unintended attitude changes). 



The most relevant sections of Part 91 would be 91.3 and 91.13:

14 CFR 91.3 (FAR 91.3) says:


  §91.3   Responsibility and authority of the pilot in command.
  (a) The pilot in command of an aircraft is directly responsible for, and is the final authority as to, the operation of that aircraft.
  . . . . 


and 14 CFR 91.13 (FAR 91.13) says:  


  §91.13   Careless or reckless operation.
  (a) Aircraft operations for the purpose of air navigation.
   No person may operate an aircraft in a careless or reckless manner so as to endanger the life or property of another.  
  
  (b) Aircraft operations other than for the purpose of air navigation.
  No person may operate an aircraft, other than for the purpose of air navigation, on any part of the surface of an airport used by aircraft for air commerce (including areas used by those aircraft for receiving or discharging persons or cargo), in a careless or reckless manner so as to endanger the life or property of another.


So basically it's perfectly legal for a pilot to fly with their arms crossed and their feet flat on the floor, even without an autopilot, as long as the pilot in command remains able to exercise operational control of the aircraft as required to ensure the flight remains safe and legal, and as long as they are not acting in a "careless or reckless manner" (like letting their altitude wander all over the place while sitting with their arms crossed).