Yes, and when your most valuable follow is touchy enough as it is, you definitely want to investigate him over hearsay accusations! That will really help their working relationship. The only reason you think this is a good move is premptive hindsight - you know it will work.

How is that relevant? No one knows Taim is a Darkfriend, and even our evidence is circumstantial. All we REALLY have is his order to the renegades to kill Rand, and that could have been for ordinary reasons (though it is unlikely). Everything else could be written off as coincidence or have innocent explanations. For example, the red and black motifs are explained by the name of the Black Tower and the contrast with the White, and the red as a thumbed nose at the Red Ajah, the way Rand got a kick out of paying people with Tar Valon coins.

He has no evidence that there IS a mess.

He was overseeing the reconquest and administration of Arad Domon. Given how people shit a brick over having to read about a character's baths, I am fairly sure the scenes of Rand doing paperwork and reading reports and whatnot were left out in favor of the scenes where he learned important stuff, or unusual things happened to him. We laugh off distances thanks to Travelling, but just because you can pop back to Arad Domon from Andor in five minutes does not mean you can actually do that when you are trying to rebuild the country. Siuan, Elaida, Egwene and Elayne can barely get away from the demands on their time to look into matters in their own HQ in person. Even if his workload is much reduced compared to theirs, Rand can't really take a few days off from the reconstruction effort to follow up on one man's jealous carping.
The fact that Rand gets everything he asks for from the Black Tower probably also reinforces the impression that Logain is blowing smoke up his ass, or else the issues Logain has are much pettier and insignificant compared to the imminent end of the world. As far as Rand is concerned, as long as it doesn't interfere with his efforts at Tarmon Gaidon, the Black Tower can explode into a bloody civil war with collateral damage across the wetlands, as long as it waits until the day after the Last Battle to break out.

Because of their actions. Egwene thinks she is working for the greater good, but all of her actions center around achieving power for herself. She claims she WOULD HAVE stepped aside for Elaida to preserve unity, but the idea of Elaida punishing the rebels (for the very acts Egwene agree are wrong once she gets into Elaida's shoes) is just so wrong that she can't allow her rival to win. Thoughts - "I want unity more than I want power" Actions - Keeps power, keeps up polarizing factional struggle. And don't try to claim this is inspired by some antipathy to Egwene - I argue the same principle for every character, contrasting their actions with their thoughts.

That's just it. Rand is acting in general, but Egwene, despite most of her experience being with the White Tower, is one of the least experienced when it comes to the Tower, and her views on warfare are woefully ignorant. She has no experience or expertise to determine what any of that entails. As for Rand, he is not basing anything he does on his personal qualifications, but because he is the specific person designated by the Pattern as the sole chance to effect the necessary outcomes. The day anyone can point to a particular prophecy indicating the same thing in a way that specifically indicates Egwene as an individual which motivated her claim to power, is the day I excuse and accept the things she has done to get there. In addition, Rand's enemies and adversaries are the sort who actually justify what he does. Going to war against the Forsaken or an invading army is obviously and inherently justifiable. Going to war against someone you do not believe to be evil just because people are not obeying you in a manner you believe your 18 years and six months on the job deserves - NOT THE SAME THING.

Lots of politicians risk their lives to gain more power. That is essentially what Egwene did - she only ever resigned herself to martyrdom when the decision was in another's hands. Julius Caesar, on the day he stood for election as Pontifex Maximus (basically, the high priest of Rome), told his mother when he left the house that he would return as the pontifex or not at all. Just because someone is willing to risk death to gain a greater measure of power does not make them selfless or good. And just because not ALL power-seekers would do this does not mean anyone who does it is therefore not a power-seeker. All of Egwene's "sacrifices" were a case where she forebore to settle for a reduced measure of power, instead trying to keep the largest possible power base intact, in the hopes of claiming it all. She's like the contestants on a game show who rather than settle for a lesser amount, try for the grand prize.

Says who? Just because you cannot imagine how things could be different than she claims does not make it so. In order to prove the truth of her assertions, you need to prove, first that unity IS inherently good (a notion which I reject out of hand), and second that she does not stand to gain by this end (she DOES, in fact, stand to gain from adhering to unity, in that if she wins, she gets a greater amount of power and number of followers; all she is doing when she chooses to "sacrifice" for unity is taking a path where she is less likely to win, but if she does, will get a bigger payoff). Finally, just because Egwene thinks she is doing good, does not make it so. THAT error is a classic case of the PoV trap! Just because a tyrant genuinely believes he is doing what he does for the better, does not change what he does. Did Hitler genuinely believe his actions were in the best interests of Germany, Europe and the world as a whole, or was he maliciously chortling over the destruction he caused? Who gives a damn! The people he killed are still dead. His opinion of the matter is irrelevent, as is Egwene's. It is practically a cliche to state that "No one is the villain in his own story." How is it that you have apparently never heard this?

Strawman reply. That is not what I was saying, so wait until someone MAKES that argument before ridiculing them for it. I was demonstrating that having good intentions is no proof of good works. You are recasting my position as: good intention is proof of ill ends.
You said: Egwene = X, therefore Egwene = Z, because X includes Z.
I said: X includes Y as well as Z
You are now claiming that I said: X = Y.

Learn to think already!

What reports? Gossip? Murmurings from one disgruntled underling who stands to gain from Taim's downfall? Rand has heard rumors that HE'S a Darkfriend. Why should he believe a complete stranger whom he only knows as a man who tried to usurp his own place? Why did Egwene not follow up on Halima's report that Elaida was a Darkfriend? Why did she not acceded to Romanda's demands to investigate every sister? Has Rand investigated the fact that he did not actually Cleanse the Taint? Because that's what Logain not only thinks, but tells the Sea Folk when he is supposed to be serving as Rand's official agent.

And when Rand gets handed a complete list of Darkfriends from an inside source with some credibility, and fails to act on it, then she has room to talk! You cannot seriously be comparing Logain's tattling with Verin's list!