If you haven't read this article there are spoilers so stop reading here.
Why does Harry's wand shoot flames at Voldemort during the initial broom ride?
A result of the 'Priori Incantatem' that happened in the Goblet of Fire - Harry's wand conquered Voldemort's and in doing so absorbed some of Voldemort's powers.
It then combined Voldemort's extraordinary skill and Harry's courage (which is what allowed him to force Voldemort's wand to echo, not his) when taking on its foe again - the fact that it was a different wand didn't matter, it still knew Voldemort and responded with advanced magic. Pg 570
What was the significance of Harry having Lily's eyes?
Snape, who loved Lily, but hated James, would have had a mixture of feelings when dealing with Harry. But Harry's eyes, which were a mirror image of Lily's eyes, reminded him of the woman he loved every time he looked at Harry. Thus, although he was unpleasant to Harry; he protected Harry at the crucial moments and remained loyal to Dumbledore's cause.
What was the babything in King's Cross?
- Doppelganger of baby Harry representing the form the fragment of Voldemort's soul took on when it entered him at Godric's Hollow.
- What Voldemort will become when he dies - the baby is an accumulation of the horcruxes already destroyed.
The main theory is that because Voldemort used Harry's blood to resurrect himself in Book 4, Lily's blood protection of Harry was allowed to live on inside Voldemort. Thus, while Voldemort was still alive (and thus Lily's protection still effective), Harry was 'tethered' to life and could not be killed.
Harry had become master of all three Hallows (the wand was not physically with him, nor was the stone, but he was master of all three) so that he became the 'master of death' a la the legend and thus did not die / had the choice whether to die or not.
Final duel with Voldemort - ? Why does the wand backfire?
The spell backfired, because it would not cast its spell against its own master.
What was the big revelation about Petunia?
- She loved Lily when she was younger and really wanted to go to Hogwarts
- Also when she desperately wanted to go to Hogwarts, she wrote to Dumbledore, asking him to let her in. Yet received a no because she was non-magical. Which of course added to the strain on hers and Lily's relationship.
Who was the person that managed to perform magic late in life?
- JK said in her FAQ/interview that she "changed her mind" by the time of writing POA, and decided not to put this character in.
What is beyond the veil in the Department of Mysteries?
- The veil was door way was straight into death.
Why do some people become ghosts and others don't?
Some people, like Nearly Headless Nick, died whilst having a fear of the life after death, and therefore choose an imitation of life - becoming a ghost - rather than going on. It is not clear whether this decision to 'go on' is made in a similar setting and fashion to Harry's sequence in 'King's Cross', however it is believed that there comes a point at which the choice is made, (whether this choice is conscious or subconscious is open to debate.)
Why does Harry not feel the Cruciatus curse when he comes back to life?
Maybe because the wand isn't working like it should, as Harry is the true master of the elder wand.
It has something to do with the same reason the Voldemort's spells aren't working on anyone. The fact that Harry dies a sacrificial death à la Lily.
Why did Voldemort pass out after he tries to kill Harry in the Forest?
Voldermort feels when each part of his soul 'dies'. The film portrays it as him feeling weaker and weaker each time. This is nearly the final part which needs to die for him to become mortal again. He passes out from the sheer force of that part of him dying.
King's Cross chapter explanation
We have the Avada Kedavra curse which would seem to wipe the soul contents of a body out. The theory is that using Harry's blood in his resurrection allows Lord Voldemort to perform this curse on Harry, without the spell rebounding again, as it did the last time he attempted it (at Lily and James' house, when Harry was still a baby.) So Harry and Lord Voldemort baby/part? are wiped out of Harry's body and go off to the train station which would appear to be a halfway house between life and death. This place is Harry's "party" and "inside your head" according to Dumbledore. However the autonomous knowledge of Dumbledore would appear to preclude that it is all in Harry's head.
Normally it would appear that Harry would 'get the train' and die or, if he were afraid of death, hang around and become a ghost (returning to the land of mortals, never able to experience the next life), but he is able to go back to his body due to the protection of his mother, a fact which Lord Voldemort again overlooked during the spell that resurrected him.
The fact that Lord Voldemort remains alive means Lily's protection is alive and it allows Harry to remain alive/return to life.
How did Neville get the sword?
It is stated that a true Gryffindor can pull the sword out of the hat. Like Harry did in book two. Neville would appear to be a true Gryffindor (brave, loyal etc.) and was thus able to bring the sword from wherever it was (with Griphook) and draw it out of the hat.