The opening scene gives an exact characterization of Somerset's thinking patterns and reasoning in comparison to those of the people around him. When investigating the murder-suicide in the beginning, he sees children's drawings on the fridge and asks if the child saw the deaths, much to the chagrin of the officer accompanying him. What the officer doesn't realize is that, if the child saw it, their testimony could be the difference between this being labeled a murder-suicide...or a double murder.
When Mills is chasing John Doe, he looks out of an apartment window and Doe fires on him from outside, next to some birds. One would expect the birds to take flight due to the noise from the gunshots, but one could probably conclude that even the birds in the city are apathetic to the goings on around them.
How skewered Doe's interpretation of Christianity is becomes even more apparent when you realize that most of Doe's theology comes from post-Biblical sources and even fictional literature not recognized as canonical by any church. What Biblical references he does make are fairly vague and common knowledge to a layman. It shows how little he actually knows/cares about actual Christianity, using the themes as more of an excuse than actual inspiration.
Some question Doe's plan in the end of the movie, as it appears that his goal was to kill seven practitioners of the deadly sins, but at the end one innocent victim is killed, and his chosen personification of Wrath will assuredly not be killed - at worst, he'd likely be given a minimal sentence for a crime of passion. But the case of Wrath is actually a case of Shown Their Work. During the time period when Dante lived if a man was sentenced to death they could either kill him or kill his wife and children. The loss of his entire family was considered equal to taking his life. David Mills is the Wrath victim but instead of killing him John Doe kills his wife and child.
It looks as though Somerset must have wanted Mills to kill Doe, at least on some level. If not then why does he just stand there beside Doe? If he really didn't want Mills to shoot him all he had to do was step between them and block the line of shot. But the reason Somerset doesn't stop Mills was actually already established earlier in the film. After Mills got outwitted and almost killed by John, he is very emotional and angry. Somerset tries to tell him that breaking into John Doe's apartment without a warrant would make it hard to convict him in court, since what they find could get thrown out by a judge. Mills pretends to calm down, only for him to kick the door open to John's apartment anyway. Somerset realized then that Mills is too emotional and can't be reasoned with, and therefore was as likely to shoot through him to get to Doe.
Doe's MO of turning the sin on the sinner seems to fall apart with the Sloth victim. Yes, the victim is a sinner (a drug dealing pederast according to Doe) but those crimes don't appear exactly slothful. However, the idea is that drugs allow one to escape from reality, rather than dealing with their real lives, particularly in the case of marijuana, heroin, or opiates. It's also possible Doe thought that Sloth became a drug dealer because he was too lazy to find a legitimate job.
At the press conference, the district attorney declares that the resolution of the case will be an example of "swift justice". "Justice" doesn't get much swifter than if, just hours after turning himself in, the self-confessed and proven culprit gets shot in the head.
In Orthodox Christianity, despair is classified as the eighth deadly sin, which fits Somerset, as he's become world-weary and defeatist in his law enforcement work.
Overlaps with Fridge Horror: When Mills kills John Doe, it's already been established that Doe has no legal identity or family connections. He's confirmed as a serial killer by physical evidence and confession, and just prior to his death admits he killed a man's wife to provoke him into killing him. Mills will likely see no jail time, as Doe was essentially a non-person, and only an insanely heartless judge or jury would convict him considering the circumstances; however, Doe probably arranged this on purpose so Mills will escape punishment and have no chance at any kind of penance for his wrath.
When John Doe reveals that he murdered Tracy, his exact words leading up to The Reveal are "I tried to play husband. I tried to taste the life of a simple man." This implies that before he killed Tracy he tried to do other things to her as well...
Just before John Doe turns himself in at the police station, we see him step out of a taxi. Think about it: a man covered nearly head to toe in fresh blood steps into his cab and the driver either doesn't notice or doesn't care. How jaded are people in this shitty, shitty town?
Then again, John could've gotten in the cab when it pulled over before the driver had gotten a good luck at him, put a gun to the driver's head and demanded the driver drop him off at the police station, and in exchange he'd let the driver just drive away when he got out.
It could also be that he just claimed he was injured and needed to see the police before he went to hospital.
John Doe killed Tracy, Mills' wife, shortly after Mills left for work. That means he broke into their apartment and cut off Tracy's head in either the early morning at the earliest, or mid afternoon, seeing as how it was well before dusk (7PM) when Mills and Somerset take John Doe on their little drive. One might well ask why no one reported, or at least noticed some suspicious noises or commotion in the Mills' home. The explanation? The frequent subway train traffic muffled all noise. Poor Tracy could have been screaming bloody murder at the top of her lungs, and no one would have heard her.
Doe has an obsessive hatred against "sinners", a grandiose sense of self-worth, and is a perverted sexual sadist. This makes him more guilty of Wrath, Pride and Lust than the victims of those sins. But since he's clearly a nutjob, this is probably a literal case of Insane Troll Logic.