Character mistake: Galahad claims that classifies Brogues as being 'mutually exclusive' from Oxfords. Broguing is any decorative hole punching on shoes and can be found on any shoe style.
Character mistake: When giving Eggsy a tour, Galahad tells him that Oxfords are any formal shoe with open lacing. This is incorrect as Oxfords have closed lacing.
Character mistake: Anyone who knows anything about firearms would never handle a firearm with their finger on the trigger unless ready to shoot. In the training scene with the dog Eggsy is running with his finger on the trigger the whole time. Merlin also does it in the end scene at Valentine's secret lair inside the jet.
Answer: It was to test whether the candidates were willing to do whatever it takes to get the job done. If they were willing to shoot the dog they had been caring for, it means they would be willing to make a tough choice in the field to complete a mission.