Which moments are acting on a flying airplane?
The airfoil, on its own, also has an aerodynamic moment (positive) which would make the wing rotate nose-up about the CG of the wing itself if the wing was placed free in the wind stream.


Well, that depends entirely on the location of the wing's cg. The airfoil camber determines its moment, which is negative around its aerodynamic center or neutral point for positive camber. The first derivative of that moment with respect to the angle of attack is positive, so the cambered airfoil is unstable in pitch.


  the aerodynamic moment about the aerodynamic center is constant negative


Yes. That is correct. When hinged at its quarter point, the wing will pitch down, and that moment is independent of the angle of attack if the aspect ratio of that wing is large enough.


  The physical effect is the nose up intrinsic moment of the cambered wing due to the fact that the force on the bottom and the force on the top of the wing do not act along the same line of action.


No, not really. Aerodynamic forces are pressures acting on an area, and the force is the result of the pressure difference between both sides of the wing. There is not an isolated lower and upper side force. Again, which sign the moment of the wing has depends on your choice of reference point. You only will get a nose-up moment if your reference point is aft of the center of pressure.


  For longitudinal stability, are the participating moments the moment due to Weight, the moment due to the wing lift and the moment due to the horizontal stabilizer that need to add to a zero resultant moment?


Longitudinal stability is achieved by a negative gradient of the sum of all pitching moments over angle of attack. If you balance the moments, you trim the aircraft, but that does not say anything about stability. To simplify this task, I recommend to place your point of reference into the point in which one of the forces attacks, so you can neglect this force's moment contribution. The two preferred points are the center of gravity and the neutral point of the whole aircraft, because both don't move in flight.


  Can the positive aerodynamic torque of the wing be ignored because [it is] much smaller in magnitude than the other torques?


Yes, in a first-order approximation. The tail moment is an order of magnitude bigger and dominates the pitching moments. If, however, you need to calculate the required tail size in order to trim the aircraft at all speeds, flap settings and center of gravity locations, of course you need to include the wing moment into the calculation.


  How far behind [the] CG should the wing AC be placed?


It is much easier to shift the cg than to shift the aerodynamic center around - that one comes with the wing planform. Also, for stability you should look at the distance between the whole aircraft aerodynamic center and the cg - it just happens that a cg near the wing's aerodynamic center provides the aircraft with decent stability. But the cg can very well be a bit behind the wing's aerodynamic center.


  I realized that some big airplanes have the CG behind the wing AC. In that case, the lift from the tail is force always directed upward and not downward like when the CG is in front of the wing AC.


Now you mix up center of pressure and aerodynamic center. That is one reason why I prefer to call the aerodynamic center "neutral point". The tail lift is only upwards for a rear cg location and high angle of attack, which is quite different from "always".