Can viruses that normally infect eukaryotic cells also infect bacteria? Can standard viruses infect bacteria? I'm not speaking of bacteriophage but typical RNA and DNA virus such as influenza and Ebola.

I'm not so sure about my answer. But I think "bacteriophage" is a polyphyletic name of which was defined by the ability of infecting bacteria, but not defined by the genetic, evolutionary, or morphological relationships.

It seems that if a species (or a taxon in a higher taxonomic level) of virus is infectious to bacteria, it will be classified as a member of the bacteriophage.

Here is a review article about the phage classification.