1 If you got an Amber Alert notification, it should have been with your SMS messages, as you say, unless you deleted it--or happen to be using something else, like Hangouts.  At least, the couple I've seen were there until I deleted them.
You can disable amber/weather alerts if you don't want that horrible siren thing.  It should be in the SMS settings.
I turn them off first chance I think of it when I get a new phone, because the first one I ever got was at around 3AM for something at the other end of Texas.

2

In the 2.5 years I've had this phone, this is only the second amber alert I've received, so I can't really say it's an annoyance.

The other one is gone now, too, and I'm sure I didn't delete it. It can't have been deleted for old age, either, because I've got a non-amber-alert message from Jan 2013 that's still in my history.

I think Verizon did something strange in the last update we received. This isn't the only thing that doesn't act the way it used to.

Oh, well, not really worth getting concerned about.

3 Out of curiosity, I just checked out my phone, which is running Android 5.0 Lollipop on the Verizon network.  The alert looked nothing like a text message when I got it yesterday, and is not in my text message history.  But looking into the applications folder, there is an application called Emergency Alerts, and bringing that up shows the alert.

4

We have a winner!

I looked in my apps folder and I have the same app there, and yesterday's amber alert was in it.

Thanks, David.

5 Ah, that must be a Verizon thing, then.  On Sprint, they show up in with SMS (LG G3, Lollipop, and a couple of different Samsungs from a couple years ago).

6

I think it's recent, too. They used to show up in the regular SMS list. This is Android 4.4.2; perhaps it was part of our last upgrade.

I don't think I'll be getting another upgrade. The phone model isn't being sold any longer, and there's no obvious reason for Verizon to spend the money on another one. I think the only reason we got the last one (last Autumn)  was because of the Heartbleed bug.