If you have a system of forces and you take moments about two different points will the moment be the same?
No. You can see now why dealing with torques and angular momentum is so difficult: It depends on what point in space you are taking it with respect to. The torque about one point may be different in both magnitude and direction of another arbitrary point.
If you are dealing with a system in static equilibrium however, it's a requirement that the torque about ANY point you choose is zero.
No, the moment will depend on the point. However, you do have some independence: consider the resultant force as a free vector. You are allowed to slide the point along this vector and still get the same moment.