Being aware it is not a full-fledged solution but rather a workaround (it's still unclear what's causing the underlying issue), this answer sums up what we figured being a work-around at least:
As the files/directories don't show up the usual way via "sharing the SDCard", they can still be accessed by other means. One example is using ADB: using
adb pull files can be transferred from the device to the computer, and with
adb push in the other direction. Examples:
adb pull /sdcard/Sounds/ E:\audio\phone would take the entire folder
/sdcard/Sounds and save it as
E:\audio\phone on the computer
adb push E:\audio\phone /sdcard/Sounds/ for the opposite
The culprit: types of "source" and "target" must match. If "source" is a file, "target" must be a file as well:
adb pull /sdcard/Sounds/some.mp3 E:\audio\phone would fail as the source is a file and the target a directory.
Other draw-backs can be worked around as well: if you prefer a graphical solution over tinkering with the command line, there are several good GUIs available, e.g.
And several more. As a side-effect of installing ADB, you've got a powerful tool ready which will certainly help you with many other things!
For getting ADB to work without installing the full SDK, please refer to Is there a minimal installation of ADB? More ADB related information can be found via our adb tag and its tag-wiki.
Alternatives include several "WiFi file managers", which are too many to be listed here. So if you rather want a "simple app installed" on your Android device, and then do everything with your browser, you might wish to take a look at the following lists: