Nora's tears at the end suggest, to me, genuine gratitude -- she's grateful for Kevin's belief. The spontanepus emotionality of that moment seems to deliver a clear message. He answered her vulnerability with trust. The episode begins with someone regarding Nora as a liar, which makes her indignant, and it ends with someone trusting her without hesitation, which elicits joy. I don't think Nora's position on the truth hasn't changed in-between those two moments.

She adopts a new name but, for Nora/Sarah, this could be less a lie than an affirnation of a new state of being -- having accepted her children's absence, and thinking her time with Kevin was over, she might have felt like a new person. In that sense, the new name reflects her persobal truth.

Skipping her brother's funeral is a bit hard to explain. Perhaps it's also related to her new existence as "Sarah" -- a strictly defined, re-birth of sorts that she thinks is irrevocable. Perhaps she was also evading the pain of yet another goodbye and the trauma of another lost connection. She chose, instead, to place more value on the face-to-face goodbye and did not attend the funeral. However, that face-to-face goodbye actually might have been shortly before his death. Who knows? Matt might have found her, or he may even have known where she was the whole time and they were in semi-regular contact, etc.