Voldemort never thought Draco would succeed. Voldemort gave Draco the task as punishment for Lucius's failure of getting the Prophecy.
Voldemort, seeking to punish Lucius Malfoy still further for the botched capture of Harry, demanded that Draco perform a task so difficult that he would almost certainly fail – and pay with his life. Draco was to murder Albus Dumbledore – how, Voldemort did not trouble to say.
It was a suicide mission. Voldemort assigned the task to Draco as punishment for Lucius' failure at the Department of Mysteries in OotP. He did not expect him to succeed. It's explained in the books.
Voldemort clearly does understand the rules of Wand Allegiance as he killed Snape specifically to take control of the Elder Wand, he just wasn't aware that Draco Malfoy had disarmed Dumbledore before Snape killed him.
He knew that Malfoy was behind the poison and the necklace, as he mentions this to Malfoy right before he dies: “Oh yes, I do,” said Dumbledore mildly. “You almost killed Katie Bell and Ronald Weasley. You have been trying, with increasing desperation, to kill me all year.
However, Draco was ordered by Voldemort to kill Dumbledore. Lestrange saw this as an opportunity to test Snape's loyalty and while making this vow to always watch over Draco, Lestrange forced Snape to agree to killing Dumbledore himself in case the youngster was unable to do so.
“Dumbledore didn't want to lose his wand at that point and Draco disarmed him. So that meant that the wand gave Draco its allegiance, even though Draco never knew it, even though Draco never touched it. “From that moment on, that wand gave its allegiance to Draco, and it wouldn't work as well for anyone but Draco.”
In this book, Rowling reveals that Bellatrix is the guardian of Helga Hufflepuff's cup (though she is unaware that it is a Horcrux), which Voldemort has entrusted the Lestranges to keep in their Gringotts vault.
Voldemort hugged Draco as an improvisation and it wasn't planned. Even Tom Felton (Draco Malfoy) was caught unaware which increased the effect. In Harry Potter style, it was like a welcome for Draco, but as Voldemort had never known love, he didn't know how to hug him very well. Thats why they had the awkward hug.
Dumbledore knew that destroying Voldemort's horcruxes was the only way to truly defeat him. He believed that Harry was the one person who was smart, courageous, and perhaps lucky enough to find and eliminate all the horcruxes.
Draco's refusal to reveal Harry's identity to Bellatrix was not because he liked him. It was because he believed that Harry was the only chance they had at defeating Voldemort. At first, Draco admired and revered Voldemort until he threatened his family's safety.
He finds out for sure immediately after Nagini's attack on Arthur Weasley. This night is significant in many ways. While Harry sleeps, he enters Nagini's mind.
Voldemort intentionally made six Horcruxes, but when he used Avada Kedavra on Harry, he unintentionally created a seventh Horcrux. Instead of dying, Lily's love for Harry created a counter 'curse' known as Sacrificial Protection and saved Harry.
Narcissa's greatest moment, however, was when she chose to betray Voldemort in Harry Potter and the Deathly Hallows. Not knowing whether Draco was alive or dead, Narcissa chose to lie to the Dark Lord rather than risk losing her son.
He was more jealous of Harry than he let on
Specifically, Draco was jealous of Harry. It was easy to miss because Draco didn't often show his emotions, modelling himself on his cold, confident, calculating father, but J.K. Rowling has confirmed that a lot of his enmity towards Harry stemmed from envy.
Isaacs brings up Lucius' disheveled appearance in Deathly Hallows, raising the possibility that the Malfoy patriarch had begun drinking too much. The way Isaacs saw it, Lucius was in a no-win situation. Given everything he'd done, the man no longer had a place on either side of the war.
“Harry was constantly crushing on Draco. He just couldn't hide it.” In the books and movies, Harry and Draco are constantly at each other's throats, given that Draco's parents are Voldemort supporters and the evil wizard killed Harry's parents.
J. K. Rowling stated that the pain from Harry's scar on his forehead was caused by the piece of Voldemort's soul trying to leave his body through the wound it entered to rejoin its master's soul.
She made sure HE was safe - putting THEIR LIVES at risk to save his - before they escaped. And yet while Bellatrix tortured Hermione, Draco did nothing. Because he was a weak-willed fraction of a human being who deserves nothing but contempt.
No, we don't know what spell did Molly use to take down Bellatrix, but it certainly wasn't Avada Kedavra. Whatever spell she used, it didn't disintegrate Bellatrix either.
Harry knows from his conversations with Dumbledore that if a Horcrux is in the vault, it must either be the Hufflepuff cup or else an unknown object, so they have no choice but to look for the cup.
The sword was supposed to be in her vault in Gringotts. Voldemort assigned that job to her; to keep this sword, and a few other valuable items (e.g. unknown to her, another Horcrux). So when she saw the sword in Hermione's hand, she got really scared and feared the worst. She thought her vault has been breached.
Warning The Trio To Keep Hermione Safe
At the Quidditch World Cup, Draco is quick to sneer at the Trio, and warn them to keep Hermione out of sight - as a Muggle-born, she could well be a target for the wizards rioting and tormenting the Muggle camp-owner.
The answer is simple: It was far too risky for Severus Snape to be aware of both the Hallows and the Horcruxes. You have to remember that Severus has his poker face turned on to expert level when he is employing Occlumency in the face of Voldemort, the greatest Legilimens.
With that in mind, it's best to think of Dumbledore's statement, "You are not a killer," in this way: "Draco, you are not a killer; you have not killed anyone yet, and you don't have to. The choice is before you.