A few thoughts.

The idea of Mat's Medallion going from Mat -> Elayne -> Alivia -> Gawyn seems a little too convoluted, to me. A more fitting scenario, I think, would be for Mat to give it to Elayne (as you suggested) and then for Gawyn to steal it from her (after overhearing what it does) to go confront Rand.

Yes, Alivia needs to be involved in Rand's death somehow, but to say that Gawyn looting Alivia's corpse somehow qualifies as "Alivia helping Rand die" just doesn't wash, for me.

I also think Mat's betrayal of Rand (if indeed it does occur) needs to be more blatant than this. No one would ever fault Mat (and label him a betrayer) for giving the medallion to Elayne (the mother of Rand's children, no less!), even if it did end up in Gawyn's hand sometime thereafter, and led to Rand's death. Now, if Mat were to give it directly to Gawyn, THAT would qualify (even if he didn't know the truth behind why Gawyn wanted it). But, I think the idea of Gawyn stealing the Medallion from Elayne is more plausible.

As for Alivia, I think a better scenario (to fit with your theory) would be that Rand lays dying after having been mortally wounded by Gawyn, and Alivia does something that ulimately kills him, but also (somehow) helps enable his subsequent resurrection/reincarnation (what she might do I have no idea - if Nynaeve rips him from TAR after he dies, as has been theorized before, Alivia shouldn't have to do anything to help enable that). I suppose she could just ease his passing, but I think her involvement in Rand's death needs to be more significant than that, given Min's viewing. Regardless, Alivia could then go die in her blaze of glory, as you suggested.

Lastly, I like your idea that the "Seanchan woman with the sword" from Egwene's dream is Tuon with Justice, and even the idea that Egeanin is the one to convey Justice from Rand's dead body to Tuon seems plausible enough. But there is no way that scenario has anything to do with the "who draws it out shall follow after" prophecy. The first 2 lines in that prophecy certainly refer to Rand putting callandor back into the heart of the Stone (in TSR) to hold the Tearan's hearts, and therefore the whole prophecy must refer to callandor, not Justice. I don't see how the first two lines could possibly refer to Rand stabbing Gawyn. Whose hearts would he be holding, in that case?

Anyway ... I'm just nitpicking details, here. I like the theory a lot, in general. To me, how it links to Arthurian legend is the most solid "evidence", and you did a good job tying the whole theory into the various prophecies/viewings/foretellings we know about.