Carlson is the most arrogant of the men, and the least sensitive to the feelings of others. Though he argues that it would be more humane to put Candy's old dog down, stressing that "he's all stiff with rheumatism...he ain't no good to you, Candy, an' he ain't no good to himself...why'n't' you shoot him...if you was to take him out and shoot him right in the back of the head...right there, why he'd never know what hit him", Carlson's motives are actually selfish. Candy wants the dog gone because its smell offends him; he has no sense of how Candy loves the dog, and how difficult it is to think of putting him down (Chapter 3).
Carlson's lack of sensitivity for the feelings of others are further emphasized at the end of the story, after George, knowing that Lennie is doomed, kills him to spare him from suffering he will never understand. George is understandably distraught after shooting his friend, and Slim empathetically is commiserating with him. Carlson, however, watches the two, and callously wonders, "Now what the hell ya suppose is eatin' them two guys?" (Chapter 6).
One way you might study characterization more efficiently is to color-code when you annotate the text. If you highlight the names of each of the main characters with a different color when they appear in scenes as you read, you can later go back and refer to them individually, review their actions, and develop a clearer picture of what they are like.
The fact that Carlson owns a German Luger seems intended to show that he is a World War I veteran who brought the handgun back from Europe as a souvenir. He keeps it loaded, as we see after he shoots Candy's dog. Those were tough times and tough men traveling on freight trains, living in hobo jungles, and picking up odd jobs whenever and wherever they could. Carlson keeps the gun for protection. If he was in his twenties when he served in World War I, then he would be close to forty by 1937. He sees no future for himself but hard work, low pay, and old age. He exists mainly because Steinbeck wanted to show a cross-section of the types of men who did the unskilled farm labor in the 1930s. Some were young, some were middle aged like Carlson and Slim, and some were old and used up like Candy and Crooks.
Carlson is bothered by the bad smell of Candy's old dog because it reminds him of all the bunkhouses he has had to live in for the past twenty years. It is the bad smell of his own life. He is aloof and surly. He is fed up with life and with the kind of men he has to live with. His life had a purpose when he was in the war, and he was young enough to have illusions about the future. But Steinbeck uses the incident of killing the dog for a more important reason. He had to establish that there was a gun available and that George would know how to use it when the time came to kill Lennie. When the Luger is first shown, it is what in Hollywood is called a "plant." If a weapon is going to be used at some point in a film, that weapon is customarily shown earlier to establish where and what it is. George saw where Carlson kept his German pistol and how the mechanism worked. It had to be a distinctive-looking gun. Steinbeck intended to turn the novella into a stage play immediately, and the audience would need to recognize the gun as Carlson's when George pulled it out of his pocket. They would realize that George had stolen it with the intention of killing Lennie at the place where he knew Lennie would be hiding. A German Luger is a very distinctive-looking automatic pistol. It had to be a pistol, of course, because George could hardly produce a rifle, and if he tried to strangle Lennie or even stab him to death, he might end up getting killed himself by the bigger, stronger man.
World War II changed everything for white male farm workers. The U.S. government started spending billions on ships, planes, guns, munitions, supplies, etc., even before America was drawn into the conflict. The white males were able to get good defense jobs in or near cities and never went back to the California farms again. Farm labor was taken over by men and women from Mexico, who were pretty much excluded from defense work because of the language barrier.
Carlson is typical of ranch hands who see animals more as property than pets. Carlson thinks Candy's dog smells but he also thinks the dog is useless and needs to be "put out of his misery". He doesn't truly understand that the dog is all Candy has left. Carlson still has his health and doesn't seem to have a close friend. However, even Slim seems to agree that it's time for the dog to go. After all, when Slim's dog had too many puppies to care for, Slim had no qualms about drowning some of them. Carlson thinks he is being compassionate when he offers to shoot the dog himself. However, after the dog is dead, Candy says he should have "done the job" himself. This foreshadows the ending of the novel when George must decide what to do about Lennie.