Did Boo have a bad cough at anytime in the novel?

Yes, Boo had a coughing fit after Bob Ewell attacked the children.

Still but for a man breathing heavily, breathing heavily and staggering. I thought he went to the tree and leaned against it. He coughed violently, a sobbing, boneshaking cough.

"Somebody was staggerin‘ around and pantin’ and—coughing fit to die. I thought it was Jem at first, but it didn’t sound like him, so I went lookin‘ for Jem on the ground. I thought Atticus had come to help us and had got wore out—"