Is the net force acting on an object always zero?

No because if it was, nothing would ever accelerate. If you apply a net force to an object, it accelerates. The net force acting on an object is always zero if and only if it is in static equilibrium; that is, the object is not moving (rotation or translation) or moving with constant velocity.

Given that an object is in static equilibrium, you can say that the net force and moments in all three cartesian directions are zero.