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I don't know any way of shutting down the Edison from Arduino but you can do it by opening a shell and issuing a "shutdown -P now" command.
You can also power the Edison down with the PWR button. You have to press and hold it for a while and you will see how the Edison is powered down. To power it up just press and hold the PWR button again. If you have a terminal session open while doing this process you will see how the Edison boots once it is powered up when using this method.
If I am not mistaken the card is mounted and is visible to the Intel Edison.
Normally when an SD card is seen, it is mounted so the OS can see it.
I am not sure the name of the device that was created for your SD card. However this is the general command to unmount.
umount makes sure that all data is written (flushed) to the SD card before you can remove it.
Hope this helps.