Goldfish eggs are about 3(?)mm in diameter (it's been several months since mine last layed any (that I happened to be around to see, anyway), so I'm not 100% sure on the precise size) and white-ish. They are separate from each other, but are a bit sticky, so they'll stick to plants and other surfaces (including tank walls), and may clump together.
It is entirely possible to breed goldfish successfully in a tank, however, the chances are a bit higher in a pond. Eggs in a tank won't last long if they're left with the fish.
I agree that the bubble-filled strand is most likely a gassy poop. Do you surface feed your fish? If the fish is taking in a lot of air when it feeds, it's possible for it to travel through the intestine and come out in the poop. Clear, stringy poop may also be a sign of internal bacterial infection. In any case, bubbles are not a good sign.