In terms of point 1, we can't truly diagnose asthma from standard diagnostic tests. The xray pattern we expect with asthma overlaps with numerous other diseases. Your cat's positive response to steroids does support asthma (or chronic bronchitis). An airway wash would allow for definitive diagnosis.
Cats with asthma are prone to infections in the lower airways which is one reason your vet may have been prescribing numerous antibiotics. We'd be able to isolate any bacteria by doing an airway wash and determine which antibiotics they are sensitive to.
Overall, when I look at the chronicity and see the variety of drugs used I am left to speculate that we may not be 100% certain what we are treating. For example, if your cat has asthma then you really do need to be using some sort of regular therapy (typically a steroid).
Most people aren't up for the specialist referral. We can often get things figured out without that through more of a trial and error approach. But if you're up for the referral the internist can go through their recommendations and options...