We know that Ser Davos despises Melisandre at the beginning of Game of Thrones, but now with Stannis dead and Jon Snow also, Davos suggests to take help from the Red Woman (Melisandre), so we assume that desperate situations needs desperate solutions.
How can Davos think of her without rage (hatred is not sufficient) for what happened to Shireen? Is he not aware that Shireen is not a war casualty but was already sacrificed by Stannis after suggestion from Melisandre?
Prior to S06E09: Davos doesn't know.
In the final episode of season five, "Mother's Mercy", the following happens:
The gates to Castle Black are opened and Melisandre arrives, alone and on horseback. Jon and Davos rush over to her.
JON: Stannis?
No response.
DAVOS: Shireen? The princess?
A long look but no response. Melisandre walks away from the two of them.
The GoT Wikia writes it up like this:
Davos, having returned to Castle Black at Stannis' orders, urges Jon to convince the Wildlings for aid. Jon refuses him, saying that they wouldn't side with Stannis and this is not their fight. Both of them are shocked to see a speechless and despondent Melisandre return to the Wall. Ser Davos pressures Melisandre for news, specifically asking after Shireen, but Melisandre remains silent. After seeing the empty look in her eyes, Davos lets her leave.
The very nature of Ser Davos asking if Shireen is alive is enough to indicate he has no idea what has happened to her.
IF Davos knew what had happened to her, he wouldn't be asking if she was alive or not.
Is he not aware that Shireen is not a war causality but was already sacrificed by Stannis after suggestion from Melisandre?
No. The people that would know that, were the now dead soldiers that stayed, and the sell swords & cowards that abandoned Stannis, and none of them would go to Castle Black. They went home, to the south.
The only other one would be Melisandre, who is not stupid enough to tell Ser Davos the truth. She no longer had Stannis to protect her, and she knew that Davos has no love for her already. His attachment to the princess was well know. That's why Stannis sent him away. The Red Woman did not come into power by saying stupid things.
When Davos asked, she kept silent knowing that Davos would reasonably assume that the princess died during the attack.
Davos knows, but he wants to pretend to himself that he doesn't know.
In Davos's last interview with Stannis it's clear to both Davos and Stannis that Stannis's cause is hanging by a thread, and blood magic was their best and perhaps only way of avoiding disaster. Davos has come to accept the power of blood magic, but the only royal blood around is Stannis's and Shireen's. That places Davos in an untenable position: be complicit in the death of Shireen, an innocent child who he loves, or see the defeat of his king and the triumph of the Boltons. Stannis sees Davos's dilemma, and both to spare him direct involvement and to forestall his potential interference, sends him off on a pointless mission to Castle Black. Davos perceives this, but goes along with it, and it leaves him maudlin and tongue-tied when he goes in to say goodbye to Shireen. Shireen picks up that something is amiss with Davos when he give her the carving.
He asks Melisandre only because he desperately hopes that, against all hope, Shireen was not sacrificed. Perhaps he's hoping that Stannis finally refused to kill his daughter, there was no blood magic, and that's why Stannis lost.
He probably didn't think she would be sacrificed but simply used her blood like Melisandre did with Gendry. Davos knows Gendry lived because he gave him a boat. So that's why he doesn't defy Stannis' orders but also why Stannis sends him to Castle Black. Davos would have jumped in the fire to save Shireen. Davos might think that Shireen was killed or captured by the Boltons which may be why he is so eager to help Jon defeat them - in the hopes of saving her if by chance she is being held captive or for revenge if they killed her.
He still does care about Shireen and Stannis. That is why he left to go to Castle Black, under orders of Stannis. He did not want to go but was loyal and went to go get the help of The Night's Watch. I'm sure he was suspicious of Melisandre but there is no way he knew that they were going to sacrifice Shireen. He cares for Jon Snow so much because he is the Lord Commander and needs his help. Being Ned's son might also be a part in it. Jon Snow also has the wildlings with him and needs their numbers in the wars to come. He trust Melisandre because she is a priestess of the light, which can revive people. Example, Beric and Thoros. He trust the wildlings because he trusts Jon Snow. He is still at the Wall trying to get Jon Snows help. So much stuff goes down while he's there so that's why it seems like hes been there forever. I'm sure once he finds out about Shireen from Melisandre or others, his psyche will totally change. There have been theories to what he will do when he finds out. Kill Melisandre (doubtful now that she revived Jon), March on Winterfell again with the rest of Stannis's army and The Night's Watch and save Rickon, find Gendry. All are possibilities.
TLDR He is following orders.
