The phrase outer darkness is used three times (Matthew 8:12; 22:13; 25:30). How does outer darkness in these passages relate to Hades, Gehenna and The Lake of Fire?

To answer your question, the phrase "outer darkness" is a descriptive phrase of Gehenna or the Lake of Fire. It is not relevant directly to Hades per se. The point is that while in the physical world fire is used for both light and heat, that is simply not the case with the fire of the Lake of Fire. The fire there will strictly be for the purpose of torment, but not for the purpose of providing light. Thus all of those in the Lake of Fire will be in total blackness and not see anyone else and will feel totally alone, they will simply hear the gnashing of teeth from others because of the pain. For more details on these terms and other terms, see our Messianic Bible Study MBS-107 entitled "The Place of the Dead", available in PDF, printed Manuscript or MP3 audio format.