Yes, definitely. You can add as many far field requests as you want. They can be in the same configuration, different configurations and also have different origins (and orientations).
That depends on what you mean and what you want. I'll explain below.
If you have a model with multiple antennas and you want to calculate the gain of each antenna, taking the other antennas (structures) into account, the best is to use multiple configurations. Each configuration then has a single source (for that particular antenna) and all other antennas are loaded correctly (if they need to be loaded - FEKO models an ideal voltage source and thus any system impedance would have to be added).
Regarding the location of the far filed, the origin of the far field can be changed in CADFEKO on the "Workplane" tab of the far field request. This will display the far field at that origin and the far field request will also be at that point and not at the origin. If you move the far field in POSTFEKO, as you noted, it simply changes where it is displayed and not the far field request itself.
It is also possible to calculate the far field taking only a subset of your model into account - have a look at the "Scope" tab to limit the elements to a subset. Just note that the far field will still take all the sources into account, it is just the radiation that considers only a subset of the model.
I hope this helps. It is a bit difficult to give more "to the point" answers without an example.