Before I had all the bells I could possibly desire, I always sold to Timmy and Tommy, no matter how low the amount would be! Now I will sell to whatever's closest (including the trashcan, like Lambda hahaha).
I typically use the dropbox unless it's an expensive enough fish, bug, or item. If it's one of the latter three, I'll run in and sell it in-store. I'll also sell what I have already if I just so happen to be inside the store.
I do both. It depends on what I'm doing and when. If I'm dropping off a lot, frequent visits or during the night, the box, if it's just one or two things, then them inside.