OK, lots of questions. I'll try to address some...
Elves can reincarnate.
Anything that Men don't understand how it happens, they call
magic. As for the power, it's probably the Elves' willpower and their deep connection with nature.
They can heal faster than men.
Dwarves are very good with stone, metals, etc.
I don't know about the Witch-king and Beorn, but I think that Malbeth just had the ability of his race more concentrated in him. The Numenorians in ME had foresight (eg Gilraen predicts Aragorn's future and her own, Aragorn predicts what will happen to Elrond). How - I don't know.
Elven blades are much sharpe than regular ones. Also, they can cut though "enchanted" material - eg. Sting cut through Shelob's webs. And the fact that they glow at the comming of enemies gives a big advantage to anybody who has them.
Music has a great power in the legendarium. It all begins with music - the Ainulindale. Then there is the Finrod vs Sauron singing competition. Then Luthien first singing Sauron, and then Morgoth into oblivion. In the 3rd age the song "A Elbereth Gilthoniel" gives Sam strength to continue. I'm not sure if it's the song, Elbereth's name, or just the sound of Elvish, though.
I really have no clue as to why music is so powerful. Maybe because those who know how can channel their wills through their voices or the voices of their instruments?...
I believe that there is some sort of spiit living in Old Man Willow, but I don't know how it got there.
I skipped some questions that I don't think I can do justice right now.