Well if all of her hair died at the beginning there wouldn't be much of a movie! And that was just to demonstrate what happens if it was cut.

And yes, her mother gave her the power to heal through any way possible when she drank the flower in the water. Tears, hair. Plus in the actual book, Rapunzel heals the prince with tears after her hair is cut, because the prince falls out of Rapunzel's tower and is blinded, then Rapunzel finds him and cries on him and apparently the tears are magical because he wasn't blind after that. So the directors were trying to incorporate that into the film.
Hope this helps!