lol not to gross you out but the worm could have come out of his butt or he could have dragged it in from outside. who knows. your best bet is to bring a stool sample in to the vet a week after his last worming dose.
as for the hacking, you should try to get a video of it to show you vet. its the easiest way to show things to the vet that they do at home but don't/won't do at the vets office.
Somtimes the bordatella vaccine can cause kennel cough and it does usually go away shorter after. here is a video of of dog with kennel cough, you will notice that the person can put pressure on the throat to get the dog to make the sound
here is a reverse sneeze video. you cannot make the dog produce the sound by putting pressure on the throat. usually the opposite happens.
this person is covering teh nose to make the dog breath out of his mouth instead, which will usually stop the episode. giving them a treat helps too.