Why were they supposed to shoot the puppy?

We're here to enhance your skills, test you to the limit.

Which is why you're gonna pick a puppy.

If we start from this very dialogue, the reason was quite clear. That limit can be pushed further if someone's family/friend is involved in the line of duty.

By the time training was about to end, candidates were fond of their pet, for Eggsy his pup was like a family to him. When asked to shoot his pup it was actually a matter between duty and family. To test how loyal candidate is with Kingsman. If a time comes where an agent's close one is involve in something and he is asked to do his duty, will he obey the orders or not.

The old spartan soldiers did the same thing. Bring up a dog and kill it at the end of their training. The same practice can be seen replicated in Game of Thrones. The unsullied did the dog thing too.

Even though the kill-your-dog-idea works without knowing the historical facts, it traces back to the very real idea of how to train hard, mercyless soldiers.

Also, remember that it was higher-ups that told them to shoot the dogs. They are supposed to be super soldiers/spies/agents, and should maybe just work on a need-to-know basis and listen to their superiors.

Furthermore, maybe the old English ways simply value dog lives a lot less than any human life, and to the older knights, this didn't seem as cruel as it did to modern, secular Eggsy

It's likely a combination of "how good at following our orders are you" or "do you have a problem killing something when you may not know why/feel that great about it" without hesitating.

After all, up until that point, no one had actually killed anyone yet, and as we all saw by the end of the movie, one has to be pretty good and ok with killing to be a successful kingsman.

Within the movie, this made no sense to me. We have the Kingsmen, who are self-professed honest, upright and good people, who only condone "the risking of a life to save another." And then they ask their trainees to do something arbitrarily heinous. (Afterwards they explain that of course they wouldn't risk the dog, but if that's clear, then they shouldn't be training people to do things they wouldn't consider doing.)

When you consider the needs of the plot, however, it makes complete sense. They had to bring Eggsy through the training, but at the last moment have him fail in a way that left the audience sympathetic with him, and without assigning him any explicit deficiencies (e.g. not strong enough, not a good enough marksman, etc). Having it be a high-drama moment is a bonus. Bingo: Eggsy is out, we're wondering if we trust the leader of the Kingsman, and we're all set for Eggsy to be brought back into the fold.

Sometimes the needs of the plot outweigh the needs of rationality.

I believe it's a test of trust. As a secret agent you rely on the information that it's provided to you and you must follow orders even though they don't always make sense.

It is not a test to see if they can kill when ordered no matter how attached to the target they are, it tests that the candidate trusts his commanders judgement (If I say jump, you jump).

And I also do not think that they are trying to create heartless soldiers. Their small number and the toast to the fallen agent suggest that they are more like a family than and army.

My SO had an interesting theory about this - she said that instead of looking at it as "we want you to shoot the dog", the whole point of the scene was trust. Not trusting and blindly following orders, but rather, trusting that this organization - the Kingsman - were not the bad guys, that they wouldn't allow a dog to die for the sake of a test regardless. The answer was to shoot the dog, not because you were trained to follow orders but because you were supposed to trust that the Kingsman organization wouldn't allow the callous loss of life.

This actually led me to an interesting thought of my own - the leader calls out the boy when he points the gun at him, and you hear the gunshot from the next room. He says "At least the girl had balls". This leads me to believe the correct answer might actually be to shoot the person telling you to shoot the dog.

There are three possibilities that come to my mind:

1. The gun is loaded, and they expect you to kill the dog. In this case you'd fail, but the other person probably deserves to die. Moral victory, even if it's a physical loss.

2. The gun isn't loaded, and they don't want you to kill the dog. In this case the answer is to shoot the person and explain this exact reasoning - either they did or didn't load the gun; either way the person deserves it.

3. The gun is loaded, but they don't want you to kill the dog. In this case they may be better than #1, but they still deserve to be shot for the callous action of allowing the dog to be risked for another's failure. Once again, moral victory.

We don't actually know the "right" outcome of the test (as some other answers have pointed out). We only see Eggsy fail it by failing to use the gun. Firing the gun is involved as we hear Amelia discharge her weapon. But we don't know whether she shoots the dog or her handler. It is not inconceivable that this is also a valid correct outcome (though from Galahad's subsequent explanation to Eggsy shooting the dog is at least one correct outcome).

But the cognitive dissonance about the requirement to be prepared to shoot the dog (assuming this is the intent) is not as great as some have suggested. Kingsmen are clearly meant to be prepared to do morally ambiguous things to further the cause so they have to be loyal and follow orders. They may not know the big picture so making their own judgement in critical situations is not necessarily the right thing to do. Having to shoot an innocent animal seems contrary to their moral stance. But it certainly proves their loyalty and ability follow orders.

But the setup is such that the test proves their loyalty and reliability while at the same time making a moral point. The dog doesn't die even if the agent follows the order. This tells the agents that the organisation want the agents to be reliable but doesn't waste an innocent life just to prove it. So an agent who obeys learns something very important about the Kingsman moral code despite having showed his willingness to obey orders.

In summary, the test sounds morally cruel but the outcome provides a lesson about a more positive moral stance the Kingsmen should uphold: the organisation doesn't waste life needlessly.

The correct answer is to shoot the dog. We know this because we find out later that Hart did "shoot" his dog, which he has stuffed.

As part of Allied propaganda during WWII, it was claimed that SS officers were given the same exact test. They were given a dog to raise, and ordered to kill the dog at the end of their training. This isn't true. These sort of stories are used to dehumanize the enemy.

What makes this particularly sordid and creepy however, is that following any order that contravenes the Geneva Conventions are prohibited by international law. For example, it's your duty to refuse to a commanding officer's order if the order was, for example, to torture a prisoner. This is why the defense of "I was just following orders" wasn't accepted at Nuremberg.

However, the candidates are asked to do exactly this and since Hart claimed he shot his dog, although with a blank, it's just very creepy.

I think this was possibly a test to see if they internalized the Kingsman code, for one Merlin stated once in the group training (I forget when) that:

-"Kingsman only condones the risking of a life to save another."

Given that they were truly kingsman worthy, they would assume it was a blank remembering that, along with the "This gun is live" line with no reason to assume it's true. The fact that he has to say "this gun is live" is a clear indication that it isn't, or he would of told him to load the gun himself quite possibly, just my opinion.