Android 6.0, backing up SD card before reformatting
As you state that 


  some content is visible, but I doubt that its all


I assume that by the means of tablet your card is visible and accessible. You doubts about incompleteness of displayed files are groundless, because usually most of the app data is hidden, that's why you don't see it.

I see two solutions here:


Install TWRP or any other custom recovery which allows doing full NanDroid backup.
Steps for making backup in TWRP are quite simple and described in many tutorials, e.g. this.
Gain root on the tablet (if not having already) and make manual partition backup via ADB.


After you set up ADB and gain root access, you can make backup with:

adb pull /dev/block/mmcblk0


and restore it with

pv mmcblk0 | adb shell dd of=/dev/block/mmcblk0


where mmcblk0 is the name of your card, which can be taken from the DiskInfo, for example.

You should decide which approach is simpler for you.

P.S. I highly doubt that reformatting card will fix your errors and memory issues. Their cause seems to be in installed apps, not in filesystem.