Ok, here's my preliminary list of questions.
I've only been collecting them for a couple of days.

Before the HP series -- Background stories:

What did James and Lily do for a living?
How did Molly's brothers die?
Are none of the Hogwarts teachers married? Is it a rule?
What happened to the three young muggles after Percival Dumbledore's savage attack?
What was Nymphodora Tonks' middle name?

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During -- Unanswered questions from the first six books

What happened to Voldemort's first body when the Avada Kedavra curse backfired???
What did the Dementors make Dudley see?
What are Dementors? Individual creatures? Self aware?

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Questions about the Deathly Hallows story line

Who wailed from below on page 7 of DH?
How did Aberforth get the mirror? How often did he check it?
What is a "CV" (page 34 of DH)
Why could they use port keys at Tonks' house (page 66)?
Who was the squib who learned magic later in life?
Please explain that baby-like thing whimpering in the corner.
How did Lupin and Tonks die?
How did Colin Creevey and the others die?
What did Griphook say/do when "his" sword vanished?
Did Neville's grandmother survive?
Did Umbridge get her comeuppance? Azkaban?
The ring, made by a very young Riddle, had such a terrible curse; why were the others so easy?

That Sectumsempra spell caused both George's injury and Draco's injury in HBP. Snape cured
Draco's pretty little face. Was Snape the only wizard with such knowledge?

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Questions about the epilogue -- what happened later?

What did Harry and Ginny do for a living?
What career choices did Ron, Hermione and the others make? Dudley?
How much contact did Teddy have with Harry while growing up?
Did Harry ever see the Dursleys again?
Did a relationship develope between Luna and Dean?
Did Azkaban continue to employ the Dementors?
Does the wizarding community ever learn of Voldemort's family? (A good job for that Skeeter cow.)

Did Harry, etc., go back and finish their last year at Hogwart's or were they given compensation
of some kind due to what was accomplished?...

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Got anything to add?
Would you delete any of these questions?
Other comments?