Never in TES history (or in ESO's case, future) has it been a possibility to become both a Vampire or a Werewolf.
This due to the fact that Vampirism or Porphyric Hemophilia is a disease.
Lycanthropy in Tamriel as we know it cures you of all diseases. Hence the incapability to become both.
To answer your first 2 questions in order, I recommend reading this: