Making it about a threat to others.
Then the premise might be flawed. Under the circumstances presented, using logical deduction, there had to have been a threat to others.
Not everyone is sane.
Does insanity make one's life worth less than a dog's.
No, but an insane person armed with a knife would, in and of itself, be a threat to others.
Should a person's posing a threat to oneself mean the person should be killed?
No.
If not yet a danger to others, should that person be killed?
Is this part of the premise? I didn't see this part originally.
Why would someone call the police about someone threatening a dog with a knife? If some guy wanted to kill his own dog, he could just do it himself. No need to "threaten" to do it - and no need for anyone else to even know about it. If he's threatening to kill someone else's dog, then the pet owner would have every justifiable reason to feel threatened themselves. If you were walking your dog in the park and some crazy person with a knife comes and says "I'm going to kill your dog," would you feel safe from that person? Would he not be considered a threat, since he is carrying a deadly weapon?
Now you're violating the premises of the OP with speculation.
It derails from the issue I want to discuss.
Well, as I said, the default answer would ordinarily be that human life is more important than an animal's life.
In this scenario, we have a crazy man with a knife threatening a dog. The police were called, and the police shot and killed the crazy man with a knife. Presumably, the dog's life was saved.
You're asking if he should have been killed? Probably not, if there were other ways of dealing with the situation, but without more information, it's hard to render any verdict here. I don't know what the law is on this, but it seems it would be a weak defense for a police officer to claim that he shot and killed someone to save the life of a dog. The only way it could be justified in my mind is if it could be argued that human life was in danger.