"Drag & dropping" sounds better to me, but "Dragging & dropping" has more Google results...
Which one is correct and why?
If you consider "drag and drop" as a single, atomic action, then inflecting the action as a unit makes perfect sense; you are "drag-and-dropping" your item.
If, on the other hand, the drag and the drop are two separate elements of the overall action sequence, you need to perform each element in sequence, and inflecting each verb is required; you are "dragging and (then) dropping" your item.
You'll have the same issue with other closely conjoined pairs; "cut & paste" comes immediately to mind, and "catch & release" and "smile and nod" are arguably the same.
To my mind, they are two distinct steps, as you can't perform the second without having performed the first; so inflecting each verb is how I would do it.
Because drag & drop would be considered a compound verb, dragging and dropping would be the correct way to phrase it.