At the very end of Harry Potter and the Deathly Hallows movie, Harry broke the Elder Wand and threw away the pieces. Did the Elder Wand lose its power permanently due to this? Or could someone repair it with a "Reparo" spell and use it again?
Harry broke the wand in Deathly Hallows - Part Two the movie. This is not what happened in canon.
In the book, Harry uses the Elder Wand to heal/mend his broken holly/phoenix feather wand. He then discusses the wand with Dumbledore and plans to return the wand to Dumbledore's tomb. This makes the wand vulnerable to theft. It is my guess that the Elder Wand is immune to Reparo and other basic spells like that.
So, yes, the Elder Wand could return to service again if someone stole it and began using it, or if Harry began using it.
If Harry dies a natural death, the power of the Elder Wand will be broken, as Harry, the wand's last owner, would have never lost the wand to defeat in a dual or by being disarmed.
First, in the book, the wand was intact when it went into the tomb.
Second, one cannot simply steal the Elder Wand and use its full power. For someone to take it and have complete access to its near limitless power, they would first need to defeat Harry to earn the wands loyalty, otherwise, the wand's potential is equivalent to a common wand. Harry's plan was for the wand's power to be destroyed once he naturally dies and the wands loyalty is unable to pass to another.
"I'm putting the Elder Wand back where it came from. It can stay there. If I die a natural death like Ignotus, its power will be broken, won't it? The previous master will never have been defeated. That'll be the end of it."
No. Hermione could not fix Harry's broken wand and even Ollivander said it could not be fixed. The only wand that is known to be able to fix a wand is The Elder Wand itself but that is kind of the problem because it is the Elder Wand that is broken.
“Yes,” said Harry. “Can you — ?”
“No,” whispered Ollivander. “I am sorry, very sorry, but a wand that has suffered this degree of damage cannot be repaired by any means that I know of.”