2. She makes friends quickly.
She's friends with Nynaeve, Moiraine, Elayne, Gawyn, Galad, Rand, Mat, Perrin, Amys, Bair, Melaine, Siuan and Leane. Not bad.

She considers Nynaeve and Elayne her subjects. Friendly subjects, perhaps, and they were once friends, but now Egwene has decided that she is the Amyrlin Seat, and she will have the respect she deserves!

Not arguing whether it's justified or not, just saying that they can't be considered "friends" anymore.

Moiraine tolerated her, but there was never friendship there. She "loves" Gawyn. I'm sure you've seen the criticisms of that whole situation.

Galad's nice to everyone, and besides, Egwene's pretty. They've had relatively few conversations, and certainly not enough to be called "friends." Recall that the time when they COULD have been friends was when Egwene was a novice, and thus was expressly watched to be sure she didn't get entangled with men.

Definitely not friends with Rand. Assumes the worst of him at all times, he trusts her little, she trusts him not at all. Perhaps they were friends before EotW, but since the first page she's done nothing but belittle and undermine him.

Has the same attitude towards Mat that all of the Wondergirls do.

Likes Perrin, and he's too passive to stand up to her, a fact which she takes advantage of in EotW. That was more of a "pet dog/Master" relationship than a friendly one.

The Aiel Wise Ones, I can give you. Siuan, as well. Leane is less of a friend, though, and more of a loyal retainer. She's loyal, but they're not "friends."

4. She was willing to settle for Rand.
Huh... I recall her rejecting Rand. He was devastated. For a minute. Then he started hitting on Elayne in the same chapter.

He had no desire to marry her, either. He was just worried about hurting Egwene by telling her that, because he's actually a kind person.