First off,  you haven't told us anything that makes it clear you ever had a
bought of asthma. This all could have been just allergic rhinitis. I also
don't see any reason to think it was a sinus infection (now - with 20/20
hindsight). It is that time of year and sinus infections out of the blue are
probably more common that new allergies out of the blue, especially in the
"cold and flu season", so it is a good guess except for the rest of the
history of how it relates to the dog. I'm not sure much testing was
indicated or would have done much good. Usually the intial treatment of
allergies and sinus infections is to just try the meds and then do more if
things don't go as expected. The one thing I might be a biot critical about
in your care is the continuing down the same road that wasn't working
without eliciting the additional history of allergic exposures or doing the
sinus CT to try to see why the antibiotics were failing.

Doing pulmonary function testing would help to confirm the diagnosis of
asthma but since it doesn't sound like the docs were considering that it is
understandable that they didn't order it. A CT of the sinuses might have
helped exclude a diagnosis of sinus infection but may have shown fluid in
the sinuses from the allergies as well (and isn't 100% in excluding sinus
infections either). As I said above I  might have only ordered this after
apparently adequate treatment for sinusitis failed depending on how sure it
was sinus and not allergic (or viral) it was.

I do have to wonder about a few other things. 1) Over how long were all
these visits? Sometimes sinusitis can take a few weeks to improve. If all of
the visits were in close succession the docs may have been dragging their
feet because they thought you were just not giving it enough time. 2) Was it
getting better between visits? If was improving between visits and then
relapsing, either due to absences of the dog or because they gave
antihistamines as part of the sinusitis tretment, it may have seemed that
their treatment was on the right track and that all you needed was more or
different antibiotics.

It seems pretty clear you are allergic to dogs. I'm not sure how much it
matter how much of your symptoms were from the lungs or how much from the
nose - it is clear that it was all triggered by a dog allergy (at least
fromhow you describe it). It can take longer than a few days for the allergy
symptoms to completely resolve after such a bought. The other thing is that
you probably have some dander lurking in your home. It needs to be
thoroughly cleaned. You may even need to clean the ventilation system if
general cleaning doesn't work. Taking antihistamines probably would also
help.

As for the cats - you don't seem to be allergic to them and as long as you
are not they will not be a problem. It is not a sure bet that you will
develope allergies to the cats but the history of dog allergy does put you
at risk. You'll just have to see what happens with that.