LTT in the prologue was already Mad and then cured from Madness in the prologue except he is still quite fucked up. Perhaps this has something to do with the traveling style in the prologue, for LTT due to him hurting before his suicide did not think of a more effective way to travel for he was hurting and emotional.

Now LTT in the 2nd age and the other people could access the TP in a more direct way every since the bore was widen which started the events that led to the downfall of the age of legends. Since we know Rand Al Thor somehow can learn how to channel almost instinctively the TP with very little training even though the weaves are different would it not be suprising that LTT knows how to travel via the TP prior to everyone wising up learning about the Dark One and then the male group of Aes Sedai decide to seal up the bore?

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The forsaken are mad, so first do not try to figure them out with logic. They have an own form of internal consistency but one of the plot points is the shadow causes you to take an ideal to a extreme conclusion and if you were more mindful and introspective you realize you need to restrain this absurd ideal before it gets absurd with a contrasting ideal to keep it in check.

You can figure out which of the forsaken travel in the TP way but if I recall it was Ishmael aka the crazy fanatic, and the Lanfear the crazy woman who is still in love with the man who constantly rejects her. Lanfear's plan in the early books is to make Rand Al Thor mad enough that he will abandon all reason and fall for her feminine wiles and her promises of power. Eventually the two will use the gendered Choedan Kal Sangreal and take out the dark one. This is possible for Sangreal not directly in WOT but instead are inspired via the holy grail and the the two different journeys of a spiral can partake, one form is journeying inward into your own internal world and ascending to heaven from within while the other is journeying outward and descending from heaven to earth. Remember the spirals are what you get when a wheel does not return directly to itself but instead moves forward into new possibilities, aka the world is constantly turning just like the wheel of time is constantly turning, and turns always something new but also similar to the past but not a perfect recreation. The Wheel weaves as the Wheel wills, and we are only the thread of the Pattern.

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Now I do not have much more to say besides WoT has many good ideas but does a horrible job of implementing them due to Jordan getting lost in his narrative and world building without a good idea of what to do from moment to moment.