Answers:
Yes, that would be possible. Afterall, it be beside the backing of the Vidiians that Janeway and Chakotay be cured within "Resolutions" of a virus they contracted. Seeing that HIV is also a virus i.e. characterized by subtypes base on nucleic acerbic sequences, I really don't see why they could not cure find a cure for HIV.
No, they'd probably die next to the flu.
Why the Videans? Voyager is 370 years surrounded by the adjectives. I focus by this time Earth science would enjoy eliminate a great several toxic diseases.