Many years ago after 3 bouts of flu, I was left with a persistent cough and went to the doctor thinking I would get an antibiotic to shift it. Instead he sent me for a lung function test at the hospital that confirmed that I have mild asthma. I do get a wheeze when I get a chest infection, but there is not always a wheeze present.
The cold and some allergies set it off.
Yes - there is a type of asthma called cough-variant that does NOT have any wheezing as a symptom even in bad attacks.
This does make it a little higher risk because you can have such bad symptoms and doctors sometimes don't understand how much trouble you are in because they don't hear the wheezing.
http://www.webmd.com/asthma/guide/cough-variant-asthma
The answer to your question is Yes.
It depends on how bad your asthma is.
A lung function test would confirm the diagnosis.
I presume that you have had blood tests done to make sure that you are not anaemic?
It depends upon how you define asthma. In general, wheezes are part and parcel of an asthmatic attack. I think you have to look further and get another opinion. There can be many etiologies that cause shortness of breathy, from a blood disorder (get a hematological consult) to pancreatitus to fibrosis of unknown origin.