Is murder a crime in Ankh-Morpork?

Question

On one hand, the Night Watch pursues much smaller offences. On the other hand, members of the Assassins' publicly acknowledge their murders without facing any punishment.

Answer 1

The very short answer to this is "no". The reach of the City's Watch only extends to murders carried out by non-members of the Guild of Assassins. Where a registered Assassin commits the murder (or as they would say, "inhumation") and adheres to their strict rules and procedures laid out by the Guild themselves, notably providing forewarning to the person who has been targeted for assassination and providing a receipt after killing them, the Guild member who actually does the dirty deed appears to be immune from the law.

The Archchancellor gives us an overview of the realpolitik that exists within Ankh-Morpork:

‘One: the Assassins, while deadly, are not random, and indeed are mostly a danger to one another. Assassination is only to be feared, generally speaking, by those powerful enough to have a stab, as it were, at defending themselves.’
Another little globe appeared.
‘Two: it is an article of faith with them that property is undamaged. They are invariably courteous and considerate and notoriously silent, and would never dream of inhuming their target in a public street.’
A third globe appeared.
‘Three: they are organized and therefore amenable to civic influence. Lord Vetinari is very keen on that sort of thing.’
And another globe popped into life.
‘And four: Lord Vetinari is himself a trained Assassin, majoring in stealth and poisons. I am not sure he would share your opinion. And he is a Tyrant even if he has developed tyranny to such a point of metaphysical perfection that it is a dream rather than a force. He does not have to listen to you, you see. He doesn’t even have to listen to me. He listens to the city. I don’t know how he does, but he does. And he plays it like a violin’ Ridcully paused, then went on–‘or like the most complicated game you can imagine. The city works, not perfectly, but better than it has ever done... - _{^{Unseen Academicals}}

Precisely the same principle applies to the Guild of Thieves who can rob anyone they like, as long as they haven't been robbed before. You can even take out insurance to prevent the unpleasantness from happening by paying your robbery in advance:

“Do you know he arrested the President of the Thieves’ Guild this morning?”
“What for?”
“For being president of the Thieves’ Guild, it seems.”
The sergeant looked puzzled. “Where’s the crime in that?” - _{^{Guards! Guards!}}

You may also wish to note that before the arrival of Captain (then trainee) Carrot in 'Guards! Guards!' what little law the city had was largely based on private prosecutions, thief-taking (thief with a small 't') and the rump city watch consisting of around half a dozen night-watchmen concerned with preventing drunken disorder and a few dozen day-watch who were mostly concerned with traffic movements.