[Verizon Samsung Galaxy S III] Stealth111:
I'll attempt to try to help with some of the issues here, as I have direct contact with invisiblek pretty much daily. First I will start off with a few questions, NOT poking, just asking, this also may address some previous posts as well... You have a verizon SCH-I535 correct, and not another variant? Just making sure. And to start you wiped data in TWRP (wipe to install new ROM option basically). Told it to flash your 12/07 build? Did you ADD the "opengapps 7.1 ARM Pico" build to the multiple flash option, or did you flash rom, then go back in and flash opengapps? After it rebooted, it sometimes takes a while to get the setup to actually start and display, sometimes it goes to a black screen for minutes, but be patient, it will come.. At that point in time, it will initialize sim card and be connected, as this was my experience on all 3 of my S3 phones in my household. Then proceed through setup...

Is your external card formatted as "portable storage", or is it formatted as "Adopted storage". Also what type of filesystem do you have the card formatted with? What file manager are you using to try to access your card with? There was some expected changes that might have to be worked out as the "unofficial" roms were running Selinux in permissive mode, and the "official" roms are running it in Enforcing mode. That is bound to cause some small differences in the way the 2 roms behave. Also, I need to ask why you are trying to install SuperSU on a rom that already has Root (and the ability to manage it) built into the rom? There might be some issues being caused by that in itself.

I know that personally BusyBox (I rely on it daily) will NOT install on the official for me, but installs fine on the unofficials. Also, I have had limited luck with the built in file manager. I have used ES File Explorer forever, and it continues to behave well on this rom. Also, even if you are going to nandroid back and forth between official and unofficial, it is best to do a wipe of DATA before the nandroid... As far as the play store goes, I haven't experienced any lack of connectivity. I do get very small intermittent blips where it seems like it isn't connected for a second or 2, but it is very rare, and it has always been like that as far as I can remember.. Part of the reason that is for me is because I use a network extender, and it has always been the achilles heel with CM.

Try going into TWRP, do the "wipe to install new rom", add the cm zip, and ALSO add the latest "Opengapps arm 7.1 pico or micro" (your choice) build into the zip queue so that they are flashed in succession. Let it rip, and see if that offers you any different results. After you get it initially setup, go into developer options and enable root as needed. Give it a run and see what issue are still present, if any. I am surprised to hear some of these issues, as between myself, my kids, and a handful of friends, everyone is super stoked at how well these early builds are running... I myself have less than 3 reported issues from friends that we are looking into, and they are all just cosmetic bugs so far... I will be glad to try to help in any way possible..

P.S. There have been quite a few things resolved/changed between TWRP 2.8.7.0 and 3.0.2-0, I would highly suggest not only upgrading to the latest TWRP version, but also making sure you are on the firmware versions that are mentioned in the OP.

Stealth111