Galaxy S7 Edge - Change USB Mode in Phone that Will Not Boot
Did you try ADB pull? If you know the folder your files are in, you can try to pull them out. (/sdcard is the name of Internal Storage folder) If it works you should see the file you pulled in the ADB directory. 

adb pull /sdcard/your-files


Also, this can be an issue related to the Setup Wizard app. If that is the case, the following ADB command should work.

adb shell pm hide com.google.android.setupwizard


If it works you may spot some errors and missing features, but MTP should work. You might have to factory reset or reflash (not sure if that is possible on Verizon devices) to fix the errors. 

You can try contacting Samsung and Verizon customer support, and if you prove the device ownership to them (the bill or invoice should do) they might use their debug protocols to recover your files, but I'm not sure about this as I never tried it. 

If these doesn't work, I don't see any other methods unfortunately. (@esQmo_ It is currently not possible to flash TWRP in a Verizon S7 Edge because there is no known method to unlock the bootloader)

To answer your ultimate questions:

1:
There is a way to switch USB mode from ADB, but it requires root. Also, I'm surprised that your device shows up in ADB. It shouldn't if the device is in charge only mode. My guess is that your device is in MTP mode, but it doesn't show the files because of a screen lock or something, and your OS interprets it as a CD ROM Drive. 

2:
Not at all. The only way you may have a shot at recovering this is ADB. You can't do anything in Download or Recovery modes (I think you referred to Recovery as Debug?) because it will compromise data security.

3:
The options I mentioned above are the only ones I could think of. 

Also, I suggest waiting for any other answers before resetting or reflashing, as they might have methods I missed. Good luck!