[Verizon Samsung Galaxy S III] HMkX2:
Let me double check... you unchecked the "Respect CM" box? It needs to be BLANK.

"Developer Options" is in your phone's Android Settings. Click "Settings -> About Phone -> Tap on the Build Number repeatedly very fast" until it says you've enabled Developer Options. That will enable Developer Options in the settings menu. It is at the very bottom near "About Phone". Inside there there is an option that says "Root Access" - (Disabled, Apps Only, ADB Only, Apps + ADB) [persist.sys.root_access 0, 1, 2, 3]​

To verify, going in after the script ran and changing it there didn't fix it? (You can test if it worked by opening a new terminal -- saving the one you are already root -- and seeing if you can su again.)

I just tried this on my phone, and was able to re-gain root access.

Your header is moving target. Before it was this:

00000000 eb 76 90 45 58 46 41 54 20 20 20 00 00 00 00 00 |.v.EXFAT .....|
00000010 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
*
00000040 00 00 00 00 00 00 00 00 00 78 60 07 00 00 00 00 |.........x`.....|
00000050 00 08 00 00 00 10 00 00 00 18 00 00 60 60 07 00 |............``..|
00000060 04 00 00 00 a3 74 cc 30 00 01 02 00 09 08 01 80 |.....t.0........|
00000070 46 00 00 00 00 00 00 00 f4 f4 f4 f4 f4 f4 f4 f4 |F...............|
00000080 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 |................|
*
000001f0 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 f4 1f 2c 55 aa |.............,U.|
00000200

Now it is this:

00000000 eb 76 90 45 58 46 41 54 20 20 20 00 00 00 00 00 |.v.EXFAT .....|
00000010 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000020 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000030 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000040 00 08 00 00 00 00 00 00 00 78 60 07 00 00 00 00 |.........x`.....|
00000050 00 08 00 00 00 0f 00 00 00 18 00 00 60 60 07 00 |............``..|
00000060 04 00 00 00 8c d6 8c e6 00 01 00 00 09 08 01 80 |................|
00000070 00 00 00 00 00 00 00 00 33 c9 8e d1 bc f0 7b 8e |........3.....{.|
00000080 d9 a0 fb 7d b4 7d 8b f0 ac 98 40 74 0c 48 74 0e |...}.}....@t.Ht.|
00000090 b4 0e bb 07 00 cd 10 eb ef a0 fd 7d eb e6 cd 16 |...........}....|
000000a0 cd 19 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000000b0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000000c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000000d0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000000e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000000f0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000100 0d 0a 52 65 6d 6f 76 65 20 64 69 73 6b 73 20 6f |..Remove disks o|
00000110 72 20 6f 74 68 65 72 20 6d 65 64 69 61 2e ff 0d |r other media...|
00000120 0a 44 69 73 6b 20 65 72 72 6f 72 ff 0d 0a 50 72 |.Disk error...Pr|
00000130 65 73 73 20 61 6e 79 20 6b 65 79 20 74 6f 20 72 |ess any key to r|
00000140 65 73 74 61 72 74 0d 0a 00 00 00 00 00 00 00 00 |estart..........|
00000150 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000160 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000170 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000180 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
00000190 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000001a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000001b0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ff ff |................|
000001c0 ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff |................|
000001d0 ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff |................|
000001e0 ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff |................|
000001f0 ff ff ff ff ff ff ff ff ff ff ff 00 1f 2c 55 aa |.............,U.|
00000200

Those headers are very different, telling me whatever you used to format the card changed. I would need an updated debug.txt to try to see why it is not mounting now. Also, I am more interested in the "first" formatted header, the one apparently done by the Note II. The reason being, the second/newer header

should

be working since it is in a recognized format, and the first one is apparently unknown.

Yes, moar logs.

It should be running on boot, the problem is more why it's not showing up afterward. Manually type the command "vold" after you get the card mounted, and see if that forces the card to be recognized. If you get a "command not found" error, you have a big problem.

No, sdcard0 being linked is normal. Yes, you will have to hardcode the value somewhere. I believe /storage/sdcard0 is your internal storage. The links are normal, and do not affect anything -- you just need to find the "last" link in the chain, usually /storage/sdcard1 for external storage. I do not know enough about the older ROM to tell you where it "should" be mounted. (That is why I use $SECONDARY_STORAGE variable, because it varies by phone!)

Please check out Psipherious's post HERE. He made strides to get it to work on older devices. Again, without a device in my hands, I cannot develop for it. I've been giving best-guesses at this point.