Update: successful install of CM12.1 & GAPPS
It took a couple of tries to get CM12.1 installed , but it worked on the second attempt. The first attempt froze, possibly because the screen turned off. Or it could have something to do with the battery I was using which holds little charge now (even though the phone was plugged into the charger too). That GS3 also has a power button that sticks in the 'on' position and keeps rebooting itself over and over until the button unsticks (we just whack it now). Anyways, after a couple of good whacks & popping in a good battery, CM installed and then the GAPPS installed too.
TING APN
Good thing you gave us the link for the Ting APN. CM12.1 seems to have Ting listed in the APN settings (maybe for roaming?), but selecting that didn't work. I booted back into TWRP and installed the Ting APN. It still didn't work when I booted the phone to CM. I had to go to the APN settings and tap "reset to defaults" and then the long list of APNs was replaced with maybe three APNs and then the phone worked with Ting again.
THAT FREEDOMPOP BUSINESS
When we first got the phone, it had been modded to work with FreedomPop. We tried that service out, but the voice (VOIP) service didn't work well, so we cancelled that account & I can't remember if there were more steps to get it ready to work with Ting. Whatever it was, it was easy enough & we activated the phone with Ting.
Later on, we upgraded phones and deactivated this GS3. Then we activated it again as a backup phone, no problem. But I decided to do a factory (hard) reset to clear out the data, and the result of that was that it was back in a state where it only wanted to work with FreedomPop again, even though it had just been reactivated with Ting. I think the choices were to either sign up with FreedomPop again or to flash the phone with a normal ROM and do a carrier reset. For some reason, dialing ##72786# didn't cause a carrier reset, but using the Ting APN zip file worked.
LOST ROOT?
For some reason, Flashify reports that I've lost root. Is there something I need to do after flashing CM12.1 to get root status again?
Once again thanks for all your help, ZipAddict! Maybe down the road, I'll try to get better battery life set up for it by using Dkp, LSpeed, and/or XPosed.