Brother Dc:
Here's the thing. The spots may be stains on the surface, which you may be able to reduce with a surface abrasive (like rubbing compound), depending how deep they extend. Or the spots may be the result of oxidation of the steel (rust) working from the "inside" of the paint. In that case, removing some of the surface paint by abrasion could make it worse. I'd say that you've tried a reasonable, conservative approach already and should put the bike back together, step back 15 feet, and say to yourself "Cool looking bike!" Adopt the credo of physicians and archaeologists: First do no harm. And have fun!