There may be a legitimate question here, but it isn't about whether that was a suspicious inquiry. I mean: You give five good reasons why any reasonable person would think this is suspicious. Let's stipulate: It's suspicious. Or bogus. Or whatever you want to call it. I'll call it "spam."
So the question in my mind is: Presumably, Homeaway has developed a decent system for identifying and eliminating spam. Lord knows, it must have thwarted a million spammers by now. So how did this one get through their net?
Probably they are scammers. Be extremely vigilant about clicking links that look like they are inquiries from HA / VRBO or from your e-mail web mail provider. If you click on something and see something about having to re-login or session timeout, etc. don't click, just go back in through your bookmarks.
They are very good at spoofing HA /VRBO inquiries, so I repeat, if you have to re-login or session timeout do not click!