The answer is: We do not know yet for sure because this enigma has neither been disclosed in books yet nor has it been been disclosed in Show. Aye that's the worst sort of answer but It is what it is. We gotta wait till new episodes and books to find the answer for sure (For both book and show canons)
Jon's Resurrection: You are assuming that she was asking Jon to come to life, instead of asking her God. There is no evidence for your claim here. If she were to ask Jon, why bother with very long prayer and ritual? If on a cold morning my car does not start and I say "Please start", does it mean that I am asking the car to start on its own accord? It was just a way of expressing her dismay when it did not work as she had planned. Besides, Mel has already confirmed that she does not have the power to do it whenever she wants, R'hllor grants it to her if he wants to.
Night's King Raising Armies of Dead: We do not know much about the others. According to Melisandre, they are spawns of the Great other, the god of darkness whose name must not be uttered. If R'hllor (Lord of light) can grant this gift of resurrection to his followers, I suppose his rival god, Lord of darkness, can bestow a similar gift on his followers.
The two events mentioned above never happened in books. But following two did.
Dondarrion's resurrection: Thoros of Myr did not lose his faith in his god. He never had much faith to begin with. He just followed a ritual following Beric's first death, which he had learnt as a red priest. To his own surprise, Lord of Light gave Beric back. Thoros also agrees he does not have the power, only his God decides who still has some work to do for him and thus sends the dead back.
From A Storm of Swords, Chapter 39, Arya's POV:
“Could you bring back a man without a head?” Arya asked. “Just the
once, not six times. Could you?”
“I have no magic, child. Only prayers. That first time, his
lordship had a hole right through him and blood in his mouth, I knew
there was no hope. So when his poor torn chest stopped moving, I gave
him the good god’s own kiss to send him on his way. I filled my
mouth with fire and breathed the flames inside him, down his throat to
lungs and heart and soul. The last kiss it is called, and many a time
I saw the old priests bestow it on the Lord’s servants as they died.
I had given it a time or two myself, as all priests must. But never
before had I felt a dead man shudder as the fire filled him, nor seen
his eyes come open. It was not me who raised him, my lady. It was the
Lord. R’hllor is not done with him yet. Life is warmth, and warmth
is fire, and fire is God’s and God’s alone.”
Mountain's resurrection: Qyburn was stripped of his chain because he dabbled in study of necromancy and vivisection. But Qyburn learnt enough of the art to reanimate The Mountain. Qyburn's knowledge appears to be a combination of both magic and science. We can not say if the Mountain ever died in the first place. Qyburn might have found a way to keep him alive even after his head was sent to Dorne. Cersei noted that even at that time Qyburn was still experimenting and some foul thing was screaming in his chambers.
There are different sorts of magic in World of Ice and Fire. The Red Priests have their own magic, Children of Forest have their own magic, Valyrians had their own blood magic, Warlocks have their own magic etc etc. All these different types do not necessarily lead to one common source. We do not even know if gods are real in WOIAF or not but one thing is clear; the powers are real enough.