As for Melisandre and Arya, I think many on here are missing the actual context around that moment.
Melisandre, Thoros, Beric, Stannis, Jon, even the Hound; all these characters saw actual miracles performed by the Lord of Light. They didn't know what all of them meant (Mel definitely tried more than the others to interpret them) but they knew they were receiving these miracles for a reason. Stannis was meant to save Jon from the wildling attack on the wall. Thoros was to keep Beric alive long enough to help in the final battle. Jon was brought back to be able to unite all these armies together to battle the WWs. Ultimately, Beric and the Hound (probably long before he realized it) were meant to keep Arya alive. Only when Melisandre sees Arya on the wall at Winterfell, does she realize the significance of her vision looking into Arya's face back in season 3. Melisandre DIDN'T proclaim Arya the PWIP, but simply recognized that the LotL did in fact have a part for her to play as well. The pep talk she gives Arya reminds Arya what she heard in the strategy session, that the only way to win isn't simply survive, but to kill the Night King (remember, Mel wasn't there for the plan). And she knows exactly where he will be. She's killed people with blue eyes before (the waif had blue eyes) so it's not like Melisandre was calling her shot back in season 3. It was simply a reminder of the task at hand and I'm pretty sure the LotL and the God of Death (aka the Many Faced God) are rivals. Arya spent years studying under the MFG and this was completing her arch, rejecting the MFG, and returning back to being Arya Stark of Winterfell instead of "no one".
