Hi yenrou. Hopefully I can help with at least parts of this, though I don't believe I'll be able to get all of them at the moment.
I just tried with a localhost copy of WordPress with BadgeOS installed, and I agree it's not showing up in any menu at all at the moment. I'll be happy to be wrong with this, but I don't believe there's any default page that will show user achievements per user. That may very well be something that needs better documentation on.
I can say for certain that achievements/quests/your_name_here are their own custom post types, so you shouldn't be seeing them mixed in at all with the usual blog post area. However, since BadgeOS is meant to integrate as best it can with whatever theme you're currently running, it will be affected by various filters/hooks that other parts of your website use. I would wager the star ratings and social sharing parts are hooking onto the_content hook. I'd check those plugins to see if they allow selecting which post types to display on or not. Otherwise, we may need to find a workaround solution to exclude those for the BadgeOS post types. This part isn't so much a bug with BadgeOS, but just something that will need to be worked around with your setup. If you want, let me know what plugins are being used for the star ratings and whatnot and I can try to help get them to not show for your badges.