High-current motor on isolated circuit causing servos to move
If the airsoft gun thing is on a seperate circuit and galvanically isolated from the Arduino then it's possible that the arduino is getting reset/corrupted by magnetic induction. This is quite common when the wiring between the two circuits are in close proximity and the current taken by one circuit is high and pulse like.

It's the same principal for an AC transformer; the primary winding generates a voltage on the secondary due to the magnetization current in the primary. With loosely connected windings the voltage is less of course but with a bigger primary magnetization current the voltage induced gets bigger.

Here's a picture showing a bad cable method and a good cabling method: -



And another showing how fields cancel on twisted wires: -



Twisting wires that carry heavy pulses of currents (or AC) reduces the emission to other wires. Twisting wires that might be susceptible to EMI improves their ability to withstand interference. Best practise is do both.