First off, once you're infected with Corpus, you're immune to all diseases, fullstop. Presumably, it would cancel out the diseases that cause Vampirism and Lycanthropy once you're infected (but not prevent it once you've gone full-blown Vampire or Werewolf.) Outside of the Main Quest for Solstheim, there is no way for you to get the Lycanthrope disease while you have Corpus Disease.
Incubation for Werewolfism is the same as a Vampire- 3 days, then sleep.
Finally, you can have both diseases, at least in Morrowind, and they will progress. Skyrim, I think, made it so that it was impossible to be both a Vampire and a Werewolf at the same time, but it was possible to do so in Morrowind (though with a glitched backwards werewolf head.) In Morrowind, you could be infected post-change with either disease, but not with Skyrim (I think.) Not sure how Vampirism works in Skyrim, but you can't "catch" the lycanthrope disease; you can only become one by following the Companions questline.