Hi lukemccusker -- thanks for validating my problem has to do with gmail's disregard for the period in a username. Here's my story:
I have used (fictionalized) mary.smith@gmail.com as my email for years -- I always use the period for readability. I have occasionally gotten marysmith@gmail.com mail, and it's not been a problem. When I got an apple ID, I made it "mary.smith@gmail.com."
Until someone started thinking their email was marysmith@gmail.com, and is using that as her apple ID. Of course, I get all her "someone's doing something, please verify" emails, and of course I don't verify. I have no idea what her email really is -- maryasmith (or any other middle initial), or whatever. I did make contact with smithmary@gmail.com, who pleasantly looked into it, and confirmed she wasn't using marysmith inadvertently. (Our mutual name isn't nearly as common as Mary or Smith, so it's surprising.)
Today, someone (presumably the same mystery person) tried to create a Facebook account with marysmith@gmail.com, and I got that "please verify." Hopefully Facebook is more stringent than Apple about making people verify. I added the marysmith@gmail.com address to my FB account (and did verify *that* email, not the others), so hopefully she'll figure out that address is not hers.
So, lukemcc -- your solution is partial, because someone already owns marysmith@gmail.com as an AppleID. I tried to claim it/verify it by signing up for a new Apple ID, but it says it already exists as an Apple ID. I've gotten plenty of those "Your apple ID was used to sign into FaceTime on an iPhone <version I don't have>", etc., messages.
I suppose I could use the "forgot password" link to "hijack" her account, but I don't think I really want to do that.
Best answer would be for the "create a new Apple ID" process to strip periods from gmail.com IDs when it checks if a proposed ID is already being used. And if Apple support could come up with a reporting mechanism for these cases.