First a fact: RJ confirmed Rand's the rebirth of LTT's soul. One soul, two co-existing personalities and memories, two sets of skills (eg: LTT could draw really well, Rand never developped those skills) etc. You might say Rand is just a new state of existence for an entity, a soul, that previously had been LTT, and countless others before.

This is not unique to Rand, everyone carries his past lives, but normally these lives are totally isolated from the current one during incarnations.

It's not unique either to have multiple personalities/lives awake in a person. The Heroes of the Horn are aggregates of all their past personalities when dead in TAR (there is a soul pool/afterlife confirmed by RJ, so perhaps this is true for all souls as well - we don't know).

What's unique to Rand (or if Semirhage told the truth about cases in the AOL that Graendal witnessed, not unique but very rare) is that a past set of memories managed to slip through while Rand was alive.

Why? We'll never know for sure, but if it happened in the AOL and we couple that with the fact Cadsuane implied that many male channellers when they start going mad hear voices, then we can surmise a certain form of mental illness/weakness can break the "barrier", some type of schizophrenia, maybe. In Rand's case, the Taint gradually deterioated Rand's sanity, sparking something akin to schizophrenia and allowed a "personality" of LTT to invade Rand's consciousness, which lead to further mental damage (dissociation, paranoia, dark moods, weird social behaviours etc.)

Was LTT "real"? It really depends on your definition of "real". It was at the very least a construct made of partial elements of the personality and memories of the real LTT, and such "bundles" are apparently eternal within a soul, just dormant while a person (at least a Hero) is incarnated, waking up to merge seemlessly with all others when the Hero is dead and in TAR. The last personality of the soul seems to be strongest, at least if we look at the case of Birgitte. Birgitte in TAR thought of her self as Birgitte, the woman from Hawkwing's days, her last incarnation. She was conscious of her "other lives", but does not consider them as "other women" as such, it seems she is rather conscious of being a soul entity, with many incarnations, the most "real" one being her last. They form a whole, are all "her", and merged with her dominant personality (and perhaps her dominant traits which are there in every incarnation. With Rand stubborness/determination seems to be such a dominant trait of the Dragon soul). This is what Rand and LTT came to realize in TGS: that LTT's life and experiences were just a part of Rand as a soul entity, a part that ought not be disctinct as another person but merged with the rest.

It seems to me that Rand himself is responsible for "rejecting" LTT, and probably turning that part of himself into a LTT "construct", his own version of LTT as a madman (ie: he rejected the fact Ishamael healed him), a mix of the Kinslayer legends Rand knows and things about LTT Rand gleaned from the memories (his "LTT construct" became more and more complex as time passed, as Rand could no longer always himself delude himself that he was "only a madman" - the memories that came to him belied that illusion). Rand didn't want these seeping memories, he needed them to belong to someone else who wasn't him. Why? Because a lot of those memories were traumatic, and because in the third Age - and especially in the backward Two Rivers, LTT is a monster almost equal in evil to the Forsaken, a madman, a destroyer and a Kinslayer, and a suicidal man, and very importantly, a man who had seen the final devastation brought by the Shadow, that made him kill his wife and children, and lose all hope. To much trauma for a 18 y.o. to bear, especially when he discovers he must finish what this "monster" started. Rand couldn't cope with all that baggage, accept that he had all this in him, that this was part of his soul, this despair, this hardness, this guilt - and all the horrific things LTT had supposedly done (false, and sometimes real). At first, his instinct of survival in tight spots/traumatic situations made him dig inconsciously from LTT's skills/memories, to handle saidin notably (when not under big stress, Rand couldn't bring himself to dig into those memories, they were so repressed he wasn't even aware they were there). When he became conscious of that, he inconsciously started building up "LTT" as a separate personality to cope with the phenomenon - a personality distinct from his own, mad and self-destructive (Rand's repressed suicidal tendencies, present since TGH when he wanted to go to an isolated area where he wouldn't hurt anyone and die there... perhaps his first seeping memory from LTT.. perhaps just knowledge of how LTT ended it), which became more and more present as his mental state deteriorated, until he experienced by and large the same symptoms as schizophrenia (though with a totally different cause, and one that can probably be cured and was in TGS).

Through the series, Jordan frequently used Birgitte to show us the "sane" version of a soul aware of many past lives (and Mat for the humourous version), versus what was happening to Rand. Birgitte too has instincts about the Forsaken, and seing ruins or situations triggers very old memories. She is aware of that, totally unthreatened by it and for her it's all natural given who she is (though to others, like Dyelin for example, she occasionally appears as "mad" as Rand, uttering nonsenses as if she had been alive centuries ago without even realizing it. Elayne mistakes that for "arising suspicions", but more likely Dyelin just wonders if Birgitte is just a poseur thinking herself a Hero, or rather a madwoman with delusions she is Birgitte the Hero. I don,t think it has crossed Dyelin's mind as Elayne fears that Birgitte could be the real thing). For Birgitte, it's I have known the Forsaken, I have seen this city when it still stood etc., and it's more the fact these memories are fading, that she is losing parts of herself that frightens her, while Rand is afraid of losing Rand. Rand needed the distanciation from these memories, while for Birgitte they are a treasure she's losing. "It is not me, it is Lews Therin, the memories and experiences of Lews Therin, who is mad and nothing like me even though I am his rebirth", Rand coul say. When the instincts became "too strong" in a situation and Rand was repressing them/the memories, LTT began "talking" to him instead. LTT was rambling about Demandred in Rand's head when Rand faced Taim, because Rand didn't accept that he had those experiences and memories in him, that he had been LTT and his gut was telling him not to trust Taim because he's just another Barid Bel Medar, jealous of him and intent on betraying him. For those situations, Rand's mind made up a voice talking to him with those memories, and Rand could reject this as LTT's madness, preserving his illusion of control.

This evolved in a long "phase 2", where Rand was accepting the fact he had "the real" LTT in his head, and some of his memories were seeping to him, and LTT was more and more present at times. But he gradually lost control of that, more and more talking aloud in front of other people, less and less careful to hide odd knowledge, and more and more acception of the darker/hopeless/self-destructive aspects of those memories, and this got progressively worse as Rand got linked to Moridin and poisonned by his thinking, his nihilistic beliefs that tainted both Rand's behaviour and his interpretation of LTT's memories (he fed on the parts of LTT's memories that had known despair, that had lost hope).

In KOD, the shock/panic of the attack on the manor, where the only way to survive was to embrace LTT's memories (incl. memories of his skills) to a point Rand had never done, made Rand's ego sort of cower into hiding to let the "monster" LTT take over. Rand had probably reached a very dangerous spot on the path to insanity... but Semirhage soon unwittingly helped him by telling his inner circle about LTT and Rand's insanity. From there, Rand re-asserted himself (but it also lead to opening himself more and more to Moridin). Semirhage had made him confident LTT was real, and it wasn't him. It was very uncomfortable that others knew about LTT, but it was weirdly comforting to know LTT was separate from him, even if his presence threatened to drive him to final insanity. He wasn't accepting LTT as a part of himself, and his solution to force him to merge was to dominate him. He also started to request from "LTT" memories Rand himself has repressed, seemingly out of fear (pretty much everything having to do with the Sealing, most of the War and the Kinslaying... Rand has previously always refused those memories - as if he could keep the moment he'd had to face all this himself away, and most of all he repressed the fact LTT has killed his children too. Rand couldn't even touch that- and it's a solid hint that learning he was about to be a dad, before his epiphany, might have opened that wound and sent Rand into final insanity). In the meantime, Rand was more and more pushing LTT away, repressing his presence as his LTT instincts/memories were telling him more and more he was on a very dangerous path, while the more insidious presence of Moridin was driving him on that road. He was using LTT's memories selectively (and perhaps twisted in their interpretation) when it comforted him in his paranoid opinions, like his rejection of AS/Egwene, and totally rejecting him when his memories/instincts were trying to stop him (eg: when he draw the True Power, all his LTT memories were screaming "this is a disaster". By the end, Rand was saying LTT was "refusing to tell him things" and even "hiding things from him" (like, he apparently knows something important about Rand's new sword). - not surprising as Rand had almost totally repressed his LTT-memories by then, while he was more and more making himself like his delusional vision of LTT as a monster - a version of LTT that has more in common with Elan Morin Tenodrai/Moridin than the real Lews Therin Telamon, turned himself into Rand al'Thor the monster, while LTT was the man who failed, because he couldn't be strong enough, couldn't do "what must be done", frowned on things like balefiring a whole palace of people just to kill a Forsaken, when the Pattern is already so weak....

This went down to the point Rand, without the excuse of LTT's madness, nearly committed in pure rage his own Kinslaying (of Tam), and then contemplated mass murder (of the Seanchan palace in Ebou Dar, with balefire again). This was too much for Rand, he shattered and was driven to suicidal intents, a suicide of all Creation with him, the coward's way out like the Amayar.... and also the embrace of Elan Morin's beliefs that Creation is a prison for the souls, where they endlessly experience pain and misery and death, over and over.

This is when his epiphany came, his "veins of gold" to Elayne, Min and Aviendha preventing his suicide and the destruction of the world. Rand remembered love and joy, and finally he let all of LTT's memories flood him, and he became aware of Ilyena as more than a deep wound and a cause of guilt (which Rand had irrationally adopted over the death of any woman in his entourage, blaming himself when Shai'tan, the woman herslef or no one at all was responsible), understood at last why LTT had gone on and on fighting the Shadow, for love for his wife, for compassion and love for the world and humanity even if it's flawed, for life with its joys even if there's also pain - had made all these sacrifices to defeat the DO in the hope the ones he loved, himself, all humanity could go on and experience this again in future lives. Rand understood that LTT had believed this was worth it, that Creation was worth these sacrifices. The LTT after the Kinslaying had forgotten that and killed himself, but the man before the Kinslaying believed otherwise (though he had his moments of faltering, like his musing about the Creator being a gardener abandonning his garden to move on to create another). Memories of love and joys from his other past lives then flooded Rand too (a gift of the One Power who is the Creator, IMO). Rand's soul (rather than his mind, which can't be healed from the Taint according to RJ) got "healed" by a renewed faith in his role and in the Creator and his design, a reborn hope that he can succeed even if he has to die, and perhaps other loved ones will die. This is the reverse of the original prologue: an insane Rand went to DM to commit suicide, before he got healed of his darkness (by the OP/Creator instead of Shai'tan/the True Power as in the prologue) and his despair. LTT is gone, because Rand accepted that this personality was always him, a part of his own soul he had rejected as separate from him. He could let LTT "die", now that he understood Shai'tan was responsible for Ilyena's death and LTT had done what he did because he believed Creation was worth saving, for all souls, for life, for love. Ilyena died as part of the price for this victory, but LTT/Rand can comfort himself that this was the only way to offer Ilyena and the rest of humanity another chance, to avoid and end to all of this. Rand used to feed on the darkest memories and guilt of LTT, but now he's fed on his previous ones, like his love for Ilyena, his determination, his faith. So Rand has put LTT to rest, at peace at last, because he no longer sees the Dragon as a monster, no longer fears he is a monster. The LTT memories are now merged with his (and possibly much more ancient memories too, though perhaps those faded away almost immediately - we'll see in TOM and beyond), Rand doesn't need a LTT personality construct to cope with them. From now on, he'll be like Birgitte where his memories are concerned: his soul's baggage, his experiences when his soul was incarnated as LTT, part now of his incarnation as Rand al'Thor, part of his soul, his memories.

Rand also destroyed the mean by which he could put an end to Creation himself: the male CK.

The moment heralded by the Creator in EOTW has finally come to pass: "ONLY THE CHOSEN CAN DO WHAT NEEDS TO BE DONE, IF HE CHOOSES TO". On Dragonmount in TGS, Rand has finally made the final decision to be the Creator's champion and see this to the end, without faltering and killing himself, without despair, without the darkness that belongs to Moridin and the Shadow. Moridin's last gamble (since Rand wouldn't join him he might be used to destroy Creation and give Shai'tan his victory himself) has failed, and from now on it's likely Rand will turn himself into a herald of hope against Moridin's destruction of hope. Rand's ready for the Last Battle. He is the Dragon, the Champion for real now.