First, remember the mantra. RDWHAHB. Relax. Don't worry. Have a Homebrew.
As far as your over pitching is concerned, it's almost impossible to over pitch at the homebrew level, you're much more likely to under pitch. So don't worry about the over pitching at all, I probably would've thrown a whole packet in there too. Also it depends on your original gravity, recipe, etc.
Stirring the yeast is fine, as long as you had a sanitized spoon to do it with. I also assume that your wort was cooled before you pitched the yeast as if you pitched into hot wort you probably killed the cells, but assuming it was into cool wort and you used a sanitized spoon, you're fine. I sometimes stir, and sometimes just shake, when I use dry yeast.
Bucket fermenters are notorious for not getting a perfect seal, which means that the escaping CO2 can get out of any small opening easier than it can the airlock. So just because you didn't see any activity doesn't mean it's not working. I try not to open my plastic fermenters too much because you don't want to introduce oxygen, but if you're really concerned you can quickly open it and see if there is any krausen (crusty yeast stuff) on the top of the wort or if it fully fermented and dropped already, you should see residue on the side of the fermenter. If there's no activity at all, no krausen, etc, then it's possible your yeast was dead and you should pitch more if you don't see any activity at all in 72 hours. This STILL doesn't mean the batch is ruined. It's really hard to make excellent beer consistently all the time, but it's even harder to completely ruin a batch.
Once you figure out if your yeast was dead and/or if it's fermenting fine but just not bubbling, let it sit til it fully ferments. The correct way to determine if a batch is fully fermented is to take gravity readings and see when it stops dropping. But in all honesty, I never do that. I just let it sit between 3 and 4 weeks then keg or bottle it. If you see some activity (krausen, bubbles on the beer, etc) I'd either do gravity readings after a week or two, or let it sit 3-4 weeks then bottle. Taste at bottling time and assuming it tastes something like beer, let it carbonate and give it a try.
