Pullet coughing? Sneezing? Something else?

[Eggcessive]

And the "crow", is that just part of the sneeze too? It's just so weird!

Yes they can make all sorts of different sounds. I had a rooster who would sound like a bellows or pipe organ at the end of his crow. If you do a google search on chicken sneeze, look at the videos for all sorts of sounds.