I would like to lower the action on my G0, but is seems that if I go any lower, the strings will hit the bridge pickup ring. With the stock strings in place, the low e measurement at the 12th fret is 5/16. Is this normal?
Better check your measurements again. If your action is 5/16" above the 12th fret, it means that either your strings are about 1/2" above the bridge pickup, or else your neck is bowed like a banana.
Check your neck relief. Hold your low E string down at the 1st and 16th frets. Check the gap between the string and the 7th fret. It should be about .005"-.010" or approximately equal to the thickness of two pieces of paper. If its more than that, you need to tighten the truss rod to straighten the neck. This will effectively lower the strings closer to the neck and should resolve the bridge height problem that you have, as well as make the guitar much easier to play. Take it to a tech if you feel uncomfortable adjusting the truss rod.