Well now, this just gets stranger, but maybe I've at least come up with some resolution, even though it's not entirely satisfying or satisfactory.
Recall that I used Samsung Keis to back up the phone, then update it to what they had as the latest operating system version last night.
This morning, I turn the phone on, and AT&T has an update for me! I'd checked them for updates yesterday, prior to doing the Keis update, and they (AT&T) had none. Their system said I had the latest from them.
But after updating using Kies, now AT&T did have an update for me. So I let it update over the network to whatever it is that AT&T considers their latest version.
Keep in mind that AT&T (and perhaps other providers) seem to make their own versions of any release of the Android system. This is so that they can force their (AT&T's) proprietary software down your throat (you can't disable or uninstall a lot of it) and it usually has some way of getting more money from you. An example is their navigation program, which, if you elect to use it, suddenly adds about $10 per month to your phone bill. They also disable the ability to use your phone as a WiFi hotspot, even though the Android system has that feature. If you want to use that, you have to pay AT&T even more. And I'm sure that's just the tip of the iceburg.
So anyhow, AT&T doesn't want us to have the Samsung version of any OS. They want us to have the AT&T modified version. Well, I went ahead and let it update because I hoped maybe their version fixed some bug that interacts with the Google Play Services to cause the problem.
After updating, though, the phone still had crashes of Google Play Services, and when I removed it again, then I started having a LOT of battery drain from something called "media". Further, whatever was going on filled all of my phone's memory. I had 30 bytes left very soon.
So nothing would run, and there didn't seem to be any way to remedy this.
So I did a factory reset on the phone. I figured I had nothing to lose, especially since I'd backed everything up just last night.
AT&T then wanted to do yet another update of the OS! So I just let it do it to me.
Now things seem to be working fine. No crashes of Google Play Services. No excessive battery drain. Of course I'll need to set things back up the way I want, but that's not a huge deal.
But I'm finding that now, I must use "Synch" to let any of my email work. I'd studiously avoided that up until now. I do NOT want or need my phone to be "synched" with anything but itself. It's its own device. And I don't need my various email accounts merged and scrutinized by Google. I don't need my photos from one device or computer synced with anything else. I don't need texts merged with emails or phone contacts, etc. I can handle any syncing myself.
Maybe it's just a terminology thing, using this word "sync" but it is, quite frankly, ominous and frightening to me.
Big Brother is forcing us to merge things that we want to keep separate. I'm sure they want to mine all of our connections and contacts and use this for advertising and social engineering (or worse).
Anyhow, I, too, find it annoying how Google has forced this on me, and presumably billions of other people.
But the darned phone seems to be working now. If I was a conspiracy theorist, I'd think... Never mind.
Anyhow, my phone now has the following:
Model number: Samsung-SGH-1727
Android version: 4.1.2
Baseband version: 1727UCNJ3
Kernel Version: 3.0.31-844945
Build Number: JZO54K.1727UCNJ3
And my Google Play Services is:
Version: 9.2.56
And lo and behold, the App Info now shows the storage space, etc., and is not endlessly "updating". The sucker is 107 Megabytes.
And as for permissions, well, there's not much it doesn't have access to.
I took some photos of the various versions on my wife's phone, but the factory reset of my phone wiped those photos, I'm afraid. So I'll have to get it and look again later. But it may be that I now have exactly what she has.
Oh, and another interesting thing. She said that today, she picked up her phone, and found that it was in the middle of updating all sorts of Google apps, and she had NOT asked it to do so. It was just doing it on its own. She stopped it from doing it.
I don't know what to make of it all. But for now, my phone seems to be behaving nicely. But Google got what they wanted, and now I have their latest versions of everything for good or for ill!