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.
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?
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
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.