These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Alternative Character Interpretation: One could make the argument that Billy is suffering from severe PTSD and is undergoing Flashbacks instead of actually traveling through time. The Tralfamadorians could also be stated to be a non-existing coping mechanism to help deal with the trauma and the flashbacks.
Unlike most "it's all in the main character's head" interpretations, this one bears a lot of weight. After all, the Tralfamadorians look like the aliens from one of Kilgore Trout's stories, and Billy's abduction uses convoluted time-travel logic.
Additionally, the idea of being abducted by aliens and placed in a zoo was in one of Kilgore Trouts books. Relevant passage: "He got a few paragraphs into it, and then he realized that he had read it before — years ago, in the veterans' hospital. It was about an Earthling man and woman who were kidnapped by extra-terrestrials. They were put on display in a zoo on a planet called Zircon-212."
More evidence: Billy views a porn movie (or something like that... I forget the exact technology that was used) of Montana Wildhack, and that she had apparently disappeared at some point. This could have expanded into her being abducted alongside him and not coming back (in his mind).
Even more: After first being captured in the war, Billy sees one soldier being knocked around by a German soldier. The soldier asks, "Why me?" and the German replies, "Why you? Why anyone?". This same exchange is repeated when Billy is first kidnapped by the Tralfamadorians.
In addition there's the fact that Billy only starts talking about his time travel after he suffers a severe head injury. The timing of all this together really is quite suggestive.
Most importantly, let's not forget that while the novel always presents Billy's time (and space) travel in a very matter-of-fact way, as if it really were happening, it is also stated near the very opening of the story that the book is really just reporting what Billy himself says happened to him — the entire book may be told from the perspective of a hidden Unreliable Narrator.
Continuity Lockout: Despite the cameos of characters from his earlier novels, Vonnegut gives enough context for them that you don't need to have read those books to understand this one.