Tracheal collapse

When I first got Lilly, our mini schnauzer, she was only 8 weeks old (if that) and had kennel cough from being in the shelter. She coughed and sneezed for about 2 weeks (she was on meds) and then it seemed to clear up, mostly. She would cough every now and again, like she was clearing her throat, but I figured it was just a bit of left over kennel cough. It happens mainly when she runs and gets worked up.

Well, the vet came out the second time for her puppy shots and said that he thinks she may have some trachea collapse because she shouldn't really have residual coughing from kennel cough over 2 months later. He told me to monitor it to see if the symptoms persist (the other option is that because I work at the shelter, I could possibly be bringing kennel cough home on my clothes and re-contaminating her, although I doubt it) and then we would go from there. I am wondering a couple of things now:

1.) Has anyone had any experience with this?
2.) I already walk her in a harness to eliminate pressure on her throat area - anything else I can do to limit stress to that area?
3.) She is a purebred mini schnauzer - I know which pet store she came from and she was just bought there in late September or so (before she came to the shelter). I do not have her papers as the owners did not bring them in when the surrendered her. Would it be worth trying to track down the papers to see if this is a genetic problem that runs in her family, or is that too much of a crapshoot? I don't know alot in-depth about min schnauzers (although I am learning every day! [;)]) and I wasn't sure if this is something that is common among them.

Would really appreciate any advice!