Tracy Mills was murdered at the Mills household, and had her head removed and sent to Mills. Her remains should be left rotting at home by the end of the film since nobody knew of that murder yet. The Fridge Horror: the family keeps dogs indoors.
And that's assuming Doe didn't just kill the dogs to stop them from getting in the way.
Family dogs are very loyal, and so the above implications are quite unlikely indeed.
I don't care how loyal the pooches are, if they're starving and there's a hundred-plus pounds of meat just laying there...
The above statement is correct, and loyalty has nothing to do with it- there are numerous documented cases of accidental and natural death victims being partially eaten by family dogs and cats. They probably wouldn't even have any real time to rot before the pooches move in...
Doe was envious of Mills' entire life so he probably did kill the dogs.
The killer should have known that his goal to kill seven practitioners of the deadly sins was flawed. 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. Why would he set up such a statement when the statement was ultimately flawed?
Who said that all the people committing the sins had to die, or that only those people would be harmed? That's a faulty assumption on your part. It was a moral sermon on the failings of the people under the influence of each sin and a statement showing the consequences of those sins. Gluttony leads to morbid obesity/eating to death, Sloth leads to the godawful degeneration of the drug user, being guilty of Envy and Wrath led both to become murderers and take the life of an innocent or unarmed and helpless man respectively, etc.
Well, of course, he was probably never really guilty of Envy and was just manipulating Mills, but that's justified since the killer is a massive hypocrite.
The Wrath victim 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. John Doe foreshadows it when he tells Mills "whatever life I allow you to have".
Doe's plan is much more ingenious than that. To start with, he personally kills only one individual in the entire film — namely Tracy, who was innocent. In fact, all of the "sinners" are killed by someone other than Doe, and often by their own actions. Pride takes poison, Gluttony eats until he bursts, Greed mutilates himself, the Lust killer is the john, and Doe himself is shot by Mills. We don't know about Sloth but it's reasonable to assume that he was forced into the bed at gunpoint. Of course all of this is done under duress, but each person had the option of refusing to play their part in Doe's plan, and none of them did. Sloth and Mills are both still breathing when Doe is finished with them, but that is irrelevant — their lives also have been destroyed.
Except that the Gluttony victim was tied to a chair and force-fed by Doe till he burst. So Doe still kills him directly, just not via conventional means.
That's like saying Jigsaw didn't kill anyone by putting them in horrific death traps that would kill them if they didn't impart some terrible emotional or physical injury onto themselves. John Doe went into each murder planning for the death of the sinner, one way or the other. I mean, does "do this horrible thing which will eventually lead to your (or someone else's) death or I shoot you and kill you anyway" really sound like he was giving them the option to opt out, or that what they did was of their own volition? The only one you can really make a case for is the Pride murder, as he gave her a chance to get help (and people who might have been able to get her a new nose). Puppeteering people into killing themselves or others with the threat of instant death doesn't count as not committing a murder.
Of course it doesn't in real world logic, but in the mind of John Doe...
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.
He probably doesn't want to get shot either; Mills is on the edge and just as likely to shoot him anyway by accident. It might also have been instinct; if I'm not mistaken, police procedure in these matters is to keep calm and make no sudden movements which might spook the man with the gun.
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.
More generally, though: it's been raining in the city for nearly a week and it's implied the city gets a lot of rain. At the climax of the movie, our heroes head out of the city into the immediate environs... which are a flat desert that behaves as though it hasn't seen rain in monthsnote which, conveniently, is the geography closest and cheapest to Los Angeles. Right, then.
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.