Why is angular momentum of the Earth/Moon system conserved? Why is the angular momentum of the Earth/Moon system conserved, apparently unaffected by external forces such as force of the Sun?
 A: The total gravitational force of the sun acting on the earth-moon system acts through the center of mass of the system: it therefore cannot apply a torque about the center of mass, and so angular momentum about the center of mass is conserved. 
A: Angular momentum is conserved when the net external torque is zero (you're thinking about linear momentum) via $\tau = \frac{dL}{dt}$. Anyway, as long as we assume that the earth, moon, and planets are all orbiting around in a disk (which is a decent approximation), then the position vector to the Earth/Moon system from any other of the planets is in the same plane as their perturbing gravitational forces, and furthermore these other planets are sufficiently far away from the Earth/Moon system so that the position vector and force vector are essentially parallel, meaning that the torque, $\tau = \vec{r} \times \vec{F}_{Grav} $, is zero, and thus angular momentum is conserved. 
A: Actually the angular momentum of the Earth/Moon system conserved is not conserved. Considering just the earth, moon and sun in the solar system, this is the infamous three body problem for which there is no formulaic solution. The angular momentum is "mostly" conserved however.
