Sorry for the slow response. Jerry is right, but specifically, you can combine the following:
MG LED (and the other LEDs along the top edge of the board including AUX, L1, L2, and BRK) share a common +5v signal (listed as POS on the underside of the TCB board). These can be combined with +5v on the IR Receiver, +5v on the HIT LED, and +5v on the high intensity FLASH. Additionally if you are using a servo recoil you can also combine these with the + of the servo connector (middle pin of the three pin servo plug).
There are a few grounds that can be connected but not many since as Jerry says most of the switching is done on the negative side. But you can combine the IR Receiver negative (not the HIT LED negative) and the high intensity FLASH negative together. And if you are using a servo for recoil you can tie the servo negative to these as well.
Additionally if you are using the mechanical recoil/airsoft output you can combine its positive with the positive of the IR LED. This level is equal to battery voltage, not 5 volts, so only those two can be combined, but don't mix them with the positive 5 volts listed above.
The barrel elevation motor outputs can't be mixed with anything.
Anyway that should get you down to 12, but Jerry's point is worth remembering that the IR version of the slip-ring gives you a few extra wires.
Another thing to keep in mind is that the wire gauge on these slip-rings is extremely small, meaning it can't carry much current. This probably won't matter for lights and such but I could see it maybe being an issue with mechanical recoil/airsoft units since the motors in these are pretty much under-powered to begin with.
Additionally, any glitching can also cause problems with mechanical recoil/airsoft units because it can interfere with the brief signal these devices send back to the board to let it know the action has completed. Slip-rings by their nature can easily cause such glitches, so if your mechanical recoil doesn't work try by-passing the slip ring and see if that is the problem.