Does he know who she is? He knows she is a forsaken, yes but all she has done from his PoV is help him. This is especially noticeable after he calls her out in episode 5 or 6 in season 2. In their next visit she comes back to him all demure, apologetic and almost submissive. After this point whenever Rand asks for help she does. He often does not always gets to see what she does to help him only the results and god knows if Moiraine actually tells him. She takes him to see Egwene, she creates the situation to free him in Carhien, she helps take him to Falme and puts in in a position to help Egwene and kill Ishy. Of course all of these events Lanfear uses to manipulate him for her own desires. Rand logically knows Lanfear is evil but Rand also still sees the Innkeeper and eventually Mierin. He wants to believe the good person is there and that her Evil is caused be the Dark One. It is only later he realizes that Mierin was never a good person, possibly from Lews bleeding through.

And attempted to do at the beginning of tGH

Sorry for the mistake there. I was meaning to say that he should have been communicating about what was going on with Lanfear. I suppose he did tell Moiraine that she had been visiting his dreams.

I am excusing Rand for his lack of communication. Him not talking about Lanfear, even if it was to protect Egwene from Lanfear, needs to be talked about. Edit! "I am NOT excusing Rand"

Excellent Post. This also plays into Rand's arguement during their breakup in episode 6. Rand says that Egwene and he have not had a moment alone since Rhuidean, he also brings up that Egwene always drops him when she finds something more interesting. While there is a lot to break down in that statement(Egwene trying to survive lanfear attacks etc..) we will focus on one thing. It seems to be implied that Egwene has not asked about Rand's trip into Rhuidean and how he is dealing with it. I don't say this to knock Egwene, she is one of my favourite book characters but it goes to show differences between her and Rand. Rand is more empathetic wanting to know how Egwene is feeling about her recent events(Women's Circle ceremony, Seanchan Trauma etc...). Egwene is more insular and wants to deal with everything herself. Egwene tends to confide more in mentors than partners. Her relationship is Gawyn is similar. Ultimately, yes, they just do not communicate with each other at all.

While I don't think Rand is entirely blameless in his interactions with Lanfear I also don't think he is outright cheating on Egwene. Rand is clearly being groomed and manipulated by Lanfear and we cannot discount that. He has no choice in her visiting him and she is clearly more in power in the dream that him. There is also the fact that while Egwene and Rand got back together out of habit Rand clearly says they have not talked about it. Rand and Egwene's romantic interactions in season 3 are often very awkward, especially the kiss they share in episode 5. In episode 1 when they are discussing Egwene's Arc Adventures they are talking to each other but are filmed in a way to show that neither are really interacting with each other. Rafe said this was an idea from Josha and Maddy. So while they are technically together I think it is also safe to say neither are really committed to it and are only there for much needed support for recent trauma.

It is certainly Truth that the storyline in the show is going to start to deviate much much more from the books as the show progresses. Or rather parts of the storyline, depending on what they decide to cut, what they decide to rearrange and what they decide to merge.

Technically Rand cheats on Elayne in the books with Aviendha.

His thousand yard stare while talking with Egwene gave me that vibe.

She healed herself with the True Power, as she did in season 2. Some people have wondered why Lanfear was able to hold her own against Moiraine with the Chodan Ball. Well my memory is fuzzy but doesn't Rand mention that using the True power made him as or feel as powerful as when using the Chodan Kal? Lanfear's first attack on Moraine was to separate her from The Sakarnen and then decided to choke her out as the petty B would. But once Moraine had the Sakarnen back in hand Lanfear was on her back foot until she started using the True power.

Moiraine's oath to Suian, made on the Oathrod, disolved on her death is what I presume the showrunners wanted us to take from the scene.

2 Aes Sedai were holding Logain at a time, only 3(plus Moiraine) were strong enough to keep up the shield. But yeah, it is not as if Moiraine had access to the most powerful Saidar Sa'angreal to help her shield Sammael. Or other channelers to link with. It also seems she has been toying with the ability to tie off shields since she learned about it.

As for Perrin wanting to go home, it could easily be a mix of Homesickness, feeling lost after Falme and Ta'varen pull to be honest.

Quickest way to get to the Two Rivers while also avoiding the eyes of the Shadow that had just attacked them the previous night.

He was also suffering from the Longing, Perrin did mention in episode 2 that they should have gone to a stedding first." Clearly they were talking about going to a stedding in general as part of their travels.