Why does the mount point keeps changing, and how can I prevent it?

For a long-term fix, you could add the drive to /etc/fstab with a designated mountpoint. I recommend using the UUID to identify the drive.

Bonus tip:

If you want the icon to be displayed when the drive is mounted, set the mountpoint somewhere within /media/. If you don't, set the mountpoint somewhere else, such as /mnt/.

You can work around this, by unplugging the USB drive, and the going to a terminal (e.g. Application->Accessories->Terminal) and entering:

sudo rmdir /media/disk
sudo rmdir /media/disk_
sudo rmdir /media/disk__
sudo rmdir /media/disk___
etc

When you replug the disk in it should remount at /media/disk/

(Do the same for your /media/0BC7-569E)

This is not a complete answer, but rather a comment to help understand the issue better (I don't have enough reps yet to comment).

You can check diagnostic messages generated while usb / sdio devices are mounted and unmounted by running following command or by selecting (clicking on) the "dmesg" item in the left hand side of System -> Administrator -> Log File Viewer

dmesg | tail -20

The tail command prints only the most recent 20 msgs. While inserting or removing the card, run this several times or change the number at the end to see more/ less messages. This would give you an idea of what is going wrong.

Make sure you always unmount USB devices before you un-plug them.