What does the DO actually know, when you use the TP.

He is of course aware that someone is channeling a certain amount of the TP (perhaps he can distinguish the gender somehow) and likely where this generally happens in the world. Maybe he can even sense, which weave is created with the TP.

But I doubt that the DO would immediately know, who exactly the TP using person is, at least if this person never used the TP before and doesn't have special connections to the DO like the Forsaken have. I also don't think that the DO would know/see/hear what is exactly happening at the place where the TP shall be channeled; he seems too limited for that.

That means regarding the confrontation between Rand and Semirhage (if Moridin's connection wasn't used by Rand), the DO would probably just know that an unknown man is suddenly asking for the TP in the south-west. He might figure, that Rand was involved somehow. I don't think, though, that the DO directly kbew or saw that Rand was captured, tortured by Semirhage and wants to use the TP to free himself & kill Semirhage with balefire (at least if SH wasn't lurking in the Shadows, immediately communicating this info; no indication for this, I think). Being the ur-control freak he is, I normally don't see the DO giving an unknown person - let alone his arch-enemy - access to the TP for some deed he doesn't really know about, especially if there was really a plan to capture Rand.

To me it makes more sense that the DO thought that Moridin is using the TP as usual, but suddenly it flowed via their link to another person/place, and this - which never happened before - completely surprised the DO, so that Rand could use the TP for a few seconds.