It could, or it could not. SARS and the Avian Flu were predicted to be world wide pandemics by world famous scientists because global warming was going to make them worse, and we see how that turned out. But it could still happen, some day.....

Influenza has a lot of seasonal variation, which suggests that it could be affected by climate. I'm not aware of any actual research stating that influenza will become worse as a result of global warming.

No but global cooling would exacerbate it.

yes