'grats on the collection! I've recently decided to start collecting the music boxes, too, but I want to constrain it only to ones I trade in person, or find while harvesting. No auctions allowed :)
Did you find it difficult to find them all? Did you use the auctions at all?
For the DG and BB badges, I had to have the musicblox in my inventory. In both cases, the other 4 boxes were at my house, when I collected the last missing block in the wild. I didn't get the badge until I went home and added all the musicblox to my inventory. The DR badge I got with 3 at home and two collected in the wild.