1) It's not possible to get that information easily from BuddyPress right now (nor allow for things like completing only a certain group of profile fields, etc); i.e. you could hire a developer to custom build it for you. This is something that I'm going to try to make easier with future versions of BuddyPress, so plugins like Achievements can benefit :)
2) People have asked before for this, and I've put it off for various reasons. My current issue is how to make it appear (visually); and a lightbox is a good suggestion. If you or someone else are interested in doing some designs of what this could look like and how it can work, we can get it added to a future release of Achievements much quicker.
3) Badges -- they are like normal achievements, but you have to award them manually. Edit a user profile in wp-admin, scroll down to the bottom of the page, and there will be a list of all achievements you can grant or remove from a user by checking the boxes, and clicking save.