D.B.
It sounds like her immune system is depressed, so everything she is exposed to, she gets. She has no ability to fight anything off. I work in immune system support and nutritional approaches to it, and I can tell you we see this all the time. In fact, I was like your child, and one of my colleagues had a child like this for the bette part of a year - sick for 3 weeks, well for 1, sick for 3, etc. We all took proactive steps to remedy the situation, but I do understand how incredibly frustrating it is. We all did what you are doing - doctor visits, few if any answers, and mostly a whole bunch of copayments and meds to help with symptoms but of course not with the causes.
Is the long-lasting cough a deep bronchial one, from her chest (with or without coughing up phlegm) vs. a dry cough in the throat? Is it worse at night? That sounds like bronchitis - I used to get that all the time - every cold went into the chest and stayed there for 3-4 weeks. This usually has nothing to do with asthma. It made it impossible to really sleep and get any rest. If it's more in the throat, it could be related to other things including post nasal drip from her nose and sinuses (which may or may not be infected, causing the temp). I'm assuming they have ruled out strep.
The vomiting, with or without diarrhea, can also be related to her swallowing too much mucus or phlegm, or saliva from teething - but the teething doesn't explain the cough nor does it explain a temp of 103.3.
This does occur in daycare kids but it also occurs in at-home kids - it's impossible to isolate them from germs. In fact, many germs and viruses flourish in the dry air of heated homes in winter. She doesn't have the ability to fight this off so unless you do something about that, I think she will continue to pick up the next thing and the next. When you add in your soon-to-be exhausted self and a new baby, that's obviously a concern. Let me know if I can help more.