Is there any issue regarding deleting electronic files with G-d's Name in them?
Also, is there any issue with taking such a device into the bathroom?
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A digital representation of Hashem's name, whether it is His unique name or others, does not constitute writing. Consequently there is no issue of erasing it.
There is no prohibition against bringing a digital representation of Hashem's name into a bathroom per se, but to have it displayed visual would not be respectable (similar to how we don't wear a tallis gadol in a bathroom even though it's technically allowed). Furthermore we are not supposed to think about holy matters in the bathroom (except for halachos relevant to our conduct there), so it would not be appropriate to display such material in such a location.