Are there rules that require one of the pilots in a large commercial aircraft to have positive physical control of the yoke/joystick at all times?
And the same question for the rudder actuators, if any (pedals)?
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).
No. What is required by FAA is that one of the two pilots (the captain or first officer) must be the "pilot flying", and this pilot has primary responsibility for maintaining control of the aircraft at the time. The other pilot, the "pilot not flying", may assist the pilot flying by performing secondary tasks such as communication or navigation, or he/she may use the lav, have a snack/drink, read (the FAA does not prohibit reading materials not directly related to the job, but some companies do), do puzzles (ditto), etc.
Neither of these pilots are required to have hands on stick/yoke/throttle and feet on rudder pedals at all times. First off, it's redundant for most of the flight; when cruising, the plane is on autopilot, which will keep the wings level and the plane on a designated course. Second, airliners (in fact, most civilian fixed-wing aircraft) are much more stable in flight than helicopters, which do require someone to have their hands on the cyclic stick at all times to keep the helicopter "on top of" the lift vector of its rotors. So, even if the autopilot is disengaged, in normal, calm conditions the plane will almost fly itself, requiring only small corrections for turbulence. In any case, a "ham-fist" or "lead foot" on the controls is a common student mistake, much like in driving; pilots usually try to heed Han Solo's advice: "fly casual".
Nope. Sometimes they just let go and yell, "Look Ma! No hands!" into the local traffic channel. Just kidding.
There may be a slight disconnect here in how you think about pilots and the law. The average (non-pilot) person imagines that pilots are robotic servants that must obey all kinds of laws and regulations designed to "protect" passengers and slavishly follow orders from (non-pilot) FAA controllers. That is not how it works.
In the sky, the pilots are the bosses. ATC and the laws are there to help the pilots do their job better, not boss them around and force them to them to do things they don't want to do.