Question: Did David Banner get carted off in the straightjacket for murdering Bruce's mother, blowing up the lab, or for both? Did the military know about the mother? Also, where were they taking him?
Question: Is it true that Nick Nolte had problems with alcohol when making this movie?
Answer: Yes. While making the movie, he was arrested for drunk driving.
Answer: To add to the other answer, this is from Wikipedia: Nolte had struggled with substance abuse for the majority of his adult life and had begun abusing alcohol at an early age. After remaining sober for nearly 10 years, he resumed drinking in the late 1990s. Following his 2002 arrest, he again stopped drinking.
Answer: Most likely both. Since David destroyed the lab, the military went to David's house to arrest him and when they got there, they saw Bruce's mom dead so they knew he killed her. Since David was taken away in a straightjacket, they were most likely taking him to a mental hospital for a psychological evaluation to see if he was fit to stand trial or not.