Nope. Its a group effort involving Nynaeve, Siuan, Leanne, Egwene and the Wise Ones. Though Egwene is the one who finishes off Mesaana. And though I'm sure Nynaeve killed some BA, in the chaos, not too many names are mentioned.

She doesn't go away from the WT. This is one of the last scenes set there. And she isn't in Rand's last scene. So I assume she's in the Tower teaching others the new Healing as Egwene requested of her.

And she's staying because Rand asks her to convince Egwene on his plan. That may be why Nyn hasn't just up and left. Plus, Egwene asks for her support. And Nyn gives it.

Its not exactly that. As soon as she's tested, Nyn Travels outside the BT camp and knocks on Myrelle's tent. She informs Myrelle she's her equal now, and demands the bond. Myrelle asks her to come the next morning, since she's busy now (with her Warders). Nynaeve threatens to "show her the meaning of obedience", reminding her that she isn't Oath-bound yet. Myrelle gives in and transfers the bond.

As Egwene says, Nynaeve was so advanced the test was rather pointless. Nynaeve performs all the weaves with great skill, and puts them to innovative use.

The reason for doubt is the stupid requirement for calm, and the idiotic rules about walking in a measured pace, etc. Egwene had pointed out that due to Nyn's experience in TAR, she would likely be able to flout the rules like not channeling till reaching the stars. The sisters ignore this. Nynaeve flouts the rules, including running to the stars, channeling before reaching them, etc. Some AS (Lelaine, Rubinde and Barasine) get pissed. They notice that having to leave behind sick children is difficult for Nyn, so they pepper the tests with such things. Nyn is forced, on multiple occasions, to leave sick people behind, which, eventually, makes her cry and also makes her furious. Plus, there's the Balefire.

So, when she gets out, Rubinde and that crowd hold that she failed the test. Saerin is furious at the tests, saying each of those was more difficult than any she has ever seen used, and chastises them for it. Seaine and Yukiri agree. So does Egwene, but she doesn't have a say since it may look like she's favoring Nyn. Nyn argues that she was calm enough to do what had to be done, and the absolute calmness required is silly. She says if the choice was between her AS title and saving lives, she'd give up the title. Egwene approves, but Barasine is shown to be pissed off. Egwene points out that denying the shawl to the most accomplished sister among them (she implies this by pointing to Nyn's feats like cleansing Saidin, defeating Moggy, etc.). Finally, there's a vote. Saerin, Yukiri, Seaine and Romanda vote that Nynaeve has passed. Nyn refuses to swear never to use Balefire since she may have to use it at Shayol Ghul.

No problem!