My best guess would be no. The best information about how the Elixir of life works seems to come from Dumbledore:
"As for the Stone, it has been destroyed."
"Destroyed?" said Harry blankly. "But your friend - Nicolas Flamel -"
"Oh, you know about Nicolas?" said Dumbledore, sounding quite delighted. "You did do the thing properly, didn't you? Well, Nicolas and I have had a little chat and agreed it's all for the best."
"But that means he and his wife will die, won't they?"
"They have enough Elixir stored to set their affairs in order and then, yes, they will die."
Philosopher's Stone, Chapter 17, The Man With Two Faces
From this I'd surmise that the Elixir is time-sensitive. It seems to work like Felix Felicis or the Polyjuice Potion; its effects will last for so long and then eventually wear off. To benefit from immortality one has to produce Elixir through the Stone and take it at regular intervals. If you stop taking the Elixer then presumably your body will rapidly re-age. When you suddenly go from having the body of a 30 year-old to a body of (say) a 430 year-old you'd become so frail and weak that you would most likely die straight away. This seems to be what happened to the Flamels when they stopped taking the Elixir. The Elixir of Life has to be continually administered to be effective.
There is no canon information that I'm aware of that settles the question of whether the Elixir would heal you of injuries. It may work like phoenix tears, it may not. However, based on my previous paragraph, I'd suggest that if the Elixir-taker were stunned, immobilised or had the Philosopher's Stone taken away that eventually the Elixir would wear off and they would succumb to a natural death from old age. So, no, you can't throw yourself off a cliff and expect the Elixir to break your fall. The Elixir makes you immortal, not invulnerable. Similarly, if you were hit with Avada Kedavra I don't see any reason why the Elixir would protect you from a spell that has no come-back.
In short, there's no indication that the Elixir of Life has any protective qualities whatsoever.