Brief Answer:
many questions so please read the full answer...

Detailed Answer:
Hello,

for a new case of cough the infectious causes are the most likely ones. All infectious causes are potentially contagious! Some more than others but all are potentially contagious. The fever has nothing to do with the infectivity of a pathogen. Anything higher than 98.9F is usually called fever (low grade if less than 100.4F). There are special tests for influenza which may detect influenza in samples taken from the back of the nose and throat.

You can't easily distinguish bacterial from viral infections but there are some clues. This question is too broad to discuss in detail here. I can only say that there is a helpful test that may help in some cases but this is not the sole determinant of bacterial vs viral etiology. C-reactive protein in high levels is usually caused by bacteria.

The choice of antibiotic depends on the type of infection and many other factors (like your medical history, previous antibiotic exposure, etc) so there is not a single antibiotic that fits all cases.
Corticosteroids taken for a few days can be stopped cold turkey without any anticipated consequences. The same applies for antibiotics.

Your doctor can determine the source of infection (and phlegm) by examining you. There is no way that any doctor can tell where the phlegm comes from from afar.

Steroids may cause various side effects including insomnia and high blood glucose. It's OK to carry on with your physical activities if you feel up to them. During the course of infections though, the stamina is usually less than normal. Biking is not supposed to affect the phlegm, although more rapid and deep breaths may help to expel more phlegm.

I hope it helps!
Kind regards!

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