No, both Neanderthals and humans are generally believed to have each evolved from Homo habilis separately, and then later interbred to varying degrees in various locales.
Yes. And while "ancestors" is technically correct, I'm honoring my Neanderthal ancestors by calling them great^N grandparents. In other places, they had different names, but they were closely related.
Exactly. Though it's more complicated because if you go back even more generations, they're extremely distant cousins as well. The thing we're discovering about human ancestry is that it's not really a tree. Instead, it's more of a directed acyclic graph, where things branch off only to re-merge downstream.
For instance, the Denisovans are another archaic human population who split off from a common ancestor and then interbred with humans later on. And there was a recent paper which detected archaic DNA from an unknown source in West Africans. Given that species can typically interbreed for at least a million years after splitting from a common ancestor, combined with the fact that Africa had tons of hominids who coexisted with anatomically modern humans, we're probably going to discover that the story of human ancestry is highly convoluted.
Yes, it was highly convoluted. I mean, in some unregulated bordellos, non-human primates (and other species) are available. Some of which are also food animals.
Off topic, I'm guessing that your username is a reference to Battlestar Galactica. Yes?