Yes, virus or spyware can definitely affect Internet speed. Due to viruses or spyware your internet speed get down slow. You home page change automatically and redirect with another websites. Many pop ups will come when your computer with affect with viruses or spyware. Therefore you should scan your computer with updated anti virus or anti spyware removal tools.
Yes sure.
Yes sure the virus can affect the internet speed as well as the sys processing speed. You can clarify this by speed-test before virus affects what is the downloading speed & after that what is the downloading speed?