So that "de do do do de da da da" thing wasn't a stand ability, right? The Head Doctor (or Tooru) had a rock insect in his pocket and put in down in the room.
However, if the theory of HD and Tooru being two stand users is still somehow true still could mean that Tooru is the one that (possibly) inhabits the HD (with the Head Doctor being totally aware of it), controls the calamity, goes through walls, etc.
And (just a theory) maybe the HD can create rock insects out of ordinary objects? Like that stick pad thing (and perhaps a certain hair clip?). This could be possible as we see a mix of Tooru's stand and the head doctor before he puts the rock insect down. Which is so Josuke and Mamezuku can pursue it, making the HD interpret as someone pursuing HIM, as he did with the reporter guy, automatically triggering a calamity.
Of course I could be wrong and Tooru just happened to have his stand carry a rock insect on him at all times I guess.