In the book, Alex deciding he'd rather kill himself with a nice, quick, painless poison instead of his knife could be a sign that he's now unable to inflict violence on himself... or it could just be the common preference to kill oneself in a quick, painless way instead of a messy way.
The cat lady was suspicious of the knock on the door because it was similar to the setup of the assault on the writer and his wife, and told this to the police. When Alex is arrested, apparently no attempt is made to connect him to the earlier crime, since the newspapers are full of stories about the rehabilitation of the cat lady's murderer, and the writer knew who he was from those, but not because the police ever told him they'd found the man who raped his wife.
Fridge Brilliance: The government consider the writer to be a subversive; maybe they just didn't bother investigating the crimes committed against his family for that reason? From what we see of the police in Alex's society, they seem to be more hired muscle for the powerful than guardians of society.
The police didn't have enough evidence to convict him of previous crimes. Similarity of MO isn't proof. As far as the author not knowing that Alex was his own tormentor ahead of time, that's a bit of a plot hole. We might be expected to believe that teenage hooligans are so common that the author couldn't know that Alex was his own tormentor based on his crimes, but that flies in the face of the fact that the cat lady was clearly able to match Alex's MO to her own situation. Given the fact that the author has read about Alex, he should at least have an idea that Alex might be the one who killed his wife.
Remember, Alex ended up sneaking into the Cat Lady's house after the car accident ruse failed. Since the Cat Lady was killed and thus unable to testify to anything and there was obvious evidence Alex snuck in, there would be no way of them knowing Alex tried the ruse unless he told them for whatever reason. From a police standpoint, it was a simple case of a guy breaking in and killed someone.
But the cat lady already told the cops on the phone that it was similar to the earlier break in.
When Alex is released from prison, in both the book and movie, he's more or less dumped into the street and told to go away. In Real Life, people leaving prisons generally do so in a graduated way, at least in the US... they go to a "halfway house" where they re-learn how to function in the free world, and only after they've completed that are they turned loose. What did they expect when they just turned Alex loose, success or something? Even his parents didn't know that he was going to be getting out!
The government doesn't care as long as crime rates go down and they avoid bad press. That's the whole point.
In the book, they do make sure he has a place to go upon his release (as far as Alex knew, he was going home to his parents' and mentions wanting to surprise them) and even provide him with a list of job openings. Clearly they had high hopes for their treatment creating functional members of society!
More like they [would've] wanted a good outcome for the first patient so they could a.) put it into widespread use, and b.) any time it failed, say, "Well, it worked for this guy!"
Alex only got fourteen years for murder?? If Alex was fifteen in the novel and served his full sentence, he would be twenty-nine at his release. Didn't the UK have the life sentence, or are the prisons so over-crowded that judges are giving out short sentences?
In the book, the government's primary concern is with overcrowding prisons (they put 7 inmates in a cell meant for 3 at one point), so, yeah.
A 'life sentence' very rarely means literal imprisonment for life, but refers instead to the fact that the convict can never 'spend' the conviction - when he is released, it is only on parole. The judge decides how long they must be imprisoned before being considered for parole. In England, the minimum term for murder is 10 years. 14 years for a violent murder seems about right.
So the government uses Brave New World-style "deep hypnopedia" techniques to "cure" Alex of his previous "cure" while he's in the hospital. If they possess such successful hypnopedic techniques... why didn't they just use that in the first place to condition Alex against violence instead of using that Mind Rape technique?
I read the book a few years ago, but I think they mention that it's a new, untested procedure and Alex is the first guinea pig for it. I think it's implied that they eventually intend to apply it to the whole population a lá Huxley.
I believe the movie also mentions it's a new treatment.
Okay, this may be the most obvious one, but why the title, "A Clockwork Orange"?
In the book, it's the title of the novel the writer was working on. There are several explanations, you can find more on the other wiki
Alex also asks if the government vecks are trying to turn him into one after the Ludovico treatment.