I have a theory about LTT that doesn't involve time travel or torture. The DO torturing him is unlikely because LTT has been remarkably helpful in cases where the DO would definitely not want him to be competent.
Basically, we know it is possible to have memories that are not from your own life. Matt got his head filled with memories from other people, and Birgette naturally remembers memories from her previous lives (although she is now forgetting them.) In both cases, we get a perfectly well-balanced person who just happens to remember a whole bunch of stuff from the past. Note especially the fact that Birgette produces a single, integrated personality that is constant across all her lives.
I suspect that Rand is experiencing something similar, but with key differences. Unlike Matt, these memories are from his own soul. It seems likely that the soul makes a difference; give the same soul the same memories and you get the same personality back. Thus Rand's personality, as he regained his LTT memories, would begin to overlap and absorb the personality of LTT. Notice the fact that he starts unconsciously picking up LTT's mannerisms and using LTT's memories without being aware of it. Note especially that if he doesn't notice it, there is nothing crazy about what he does with LTT's memories. He is LTT reborn, and so his essence is the same person as LTT, just like Birgette reborn is always Birgette at core; a core which we see in the world of dreams.
Thus I claim that Rand has these extra memories in his head of his previous life, like Birgette. Unlike Birgette, however, he does not want to admit to himself why he has these memories. He does not want to admit that he is LTT and that he really did all the things he remembers doing. These memories are extremely painful ones for him, partly because of LTT's reputation, and especially because of what he did at the end. Combine his attempt to deny himself (because denying that he is LTT is denying himself) with fact that the taint is probably driving him crazy, and it is easy to see how he could end up constructing an alternate "persona" he could assign the LTT memories to. Thus LTT is just Rand talking to himself, refusing to admit that he's gone slightly around the bend.
Of course, part of the question is: why does he remember being LTT? The only answer I can come up with is that the taint triggered it somehow, a la Semihrage's (sp?) explanation in Knife of Dreams that crazy people (even before the taint) somehow remember past lives.