Well, I am unfamiliar with the JR ring gyro but I can tell you this much--vibration isolation is critical for performance. It looks to me like your gyro wire is stretched pretty tight and this will transfer vibration to the gyro.
Also you should not use a single piece of thick foam tape all across the bottom, but rather two narrow strips with a space between them. This will give much better vibration isolation. And make sure the case does not touch the servos, you should eliminate that piece of tape there between the gyro and the servos.

You of course have made sure the gyro is set for analog servo not digital.

I can't quite "grok" why it would cause random pulses from the main motor unless it is causing the tail motor to draw way too much current. You might play with the travel limits, I think they are set in the gyro with this one?