You don't find the gems. They're just there. If you can't see them then you're probably playing on an outdated browser. What browser/version are you using?
It's not anti-cheat measure, it's anti-cheat-breaking-the-game-and-being-reported-as-bugs-and-wasting-my-time measure. If I wanted it to be anti-cheat it wouldn't let you play instead of just being a popup :p
You can find eggs as a random event in the foothills as well. Blue eggs give penises, if you just want the penis combine them with the mushroom reagent at Ivris' shop.
Nope. Something must have cleared out local storage, and nothing in the game's code does that. A lot of "speedup" or "cleanup" programs will delete it. Things like CC Cleaner or ASC.
Yeah that error is unrelated too. It's just from pressing j to open the developer console.I have no idea what the issue is and it seems like you're the only one having it. I'd recommend trying Chrome.