The gholam tells Mat in one of their confrontations that he has a new master, who especially wants Mat dead (apparently not the Wondergirls, who the gholam hunted in ACoS/PoD as well).

I think the best guesses for this master (not mistress, btw) are either Moridin or Demandred. Graendal and Mesaana are concentrated on other plots and Moridin’s pets, Cyndane & Moghedien, probably wouldn’t be allowed to command a gholam.

One wonders, though, why Mat is the main task. While the Shadow wants the other two ta’veren dead, Mat is precisely because of his ta’veren nature, his luck & the foxhead obviously much more difficult or even impossible to kill, compared to e.g. Elayne (at least before she had the foxhead). Considerung the invasion of Caemlyn, the plot to let Rand feel pain of heart and the general will to increase chaos, it would make more sense to send the gholam – at least at first – to kill the pregnant Elayne, as an easier target (and e.g. also let the gholam free the captured DF then)?

If Demandred was the gholam’s master, if might be rather understandable, if he actually isn’t that involved with giving Rand pain of heart, but just wants to fulfill Moridin’s order to kill the ta’veren (especially another good general) if he has a chance. In my view this could indicate that it’s a bit more likely that Demandred was the gholam's master, not Moridin.

What do you think?

In this context, another theoretical question: Do you think the gholam will eventually die of thirst or hunger during his flight?

I think he mentions in PoD that blood was necessary. This would argue for the possibility that he will eventually starve.