Trivia: SPOILER! There is a mid credits scene that shows Logan going through airport security 2 years after the end of the movie, where he encounters Professor X and Magneto, who recruit him for a new mission that will be in the film X-Men: Days of Future Past.
Trivia: Guillermo del Toro originally wanted to direct the film and had talks with Hugh Jackman and the producers, but ultimately dropped out due to the massive time commitment the film required.
Trivia: Hugh Jackman wouldn't drink any liquid for a day-and-a-half before any of his shirtless scenes in order to take off any water-weight and give his physique an exaggerated "ripped" look. He said that the process was unpleasant and made him light-headed, but he was pleased with the results on-screen.
Trivia: Reportedly, both the PG-13 theatrical and unrated cut of the film were screened for test-audiences, but it was ultimately decided that the PG-13 version would be released in theaters. The unrated cut was eventually released on Blu-ray, featuring a bit more violence and more visible bloodshed.
Trivia: When Viper talks with Kenuichio on the street there is a guy behind them with a Spider-Man cap.
Answer: The healing factor was only reduced. Like you said, had it been removed he would have bled to death and every time he used his claws he would be bleeding from his hands. It is not revealed if it is at full strength at the end of the film, but it is likely it will return to full strength over time.