Hi EatRealFoo. I am not a doctor, so I can only offer general information.
I don't know if you need to be tested for H1N1 specifically. It is just a subtype of influenza type A. Often the treatments for regular seasonal flu and H1N1 swine flu are similar.
If I were in your situation and it had been this long, I would probably be asking for tests for common bacterial and viral infections, and general influenza. Sometimes they do blood tests, other times they do a throat swab, or ask you to urinate in a cup. It depends on what they are looking for.
Sometimes it takes a while for the results to come back, though. It is normal for the doctor to give you the prescription for the antibiotic and tell you to take it while you are waiting for the results, especially if you have a lot of signs of bacterial infection. Again, if it were me, I would be taking the antibiotics by now if your doctor has prescribed them.
Bronchitis is an inflammation of the bronchial tubes in the lungs. Cortisone is anti-inflammatory. It is supposed to ease the inflammation and irritation in your lungs to allow you to heal more easily.
Here is some information about cortisone, including side effects.
http://www.drugs.com/mtm/cortisone.html
Remember that most people do not have serious side effects.