Unless you have some specific thesis already in mind, I would like to suggest that you write about why John Steinbeck included that incident in his book. In my opinion, the author's intention was to show that a handgun existed, so that later on when George makes the decision to kill Lennie he will know where to find the Luger and will also know how to use it as well as where to point it at the back of Lennie's head. If you decide to go with that kind of thesis, you could quote Carlson's explanation of how he could kill the old dog painlessly with one shot. You could also quote some of Steinbeck's description of how Carlson takes the automatic pistol apart after firing it, which enables George to see exactly how to eject the clip, how to inject one shell into firing position, etc. The quote should also show that Carlson keeps the Luger under his bunk. I think it would be easier to write an essay along these lines than to try to write about Carlson's feelings, Candy's feelings, and even the feelings of the other men in the bunkhouse. You could get through that part of the incident easily by simply writing that Carlson and all the other men pressured Candy into letting Carlson kill the dog.
A German Luger is a very distinctive-looking weapon. No doubt Carlson owns a Luger because he served in World War I and brought it back from Europe as a souvenir. When George pulls it out of his pocket at the riverside rendezvous, the reader will understand immediately that George stole it from under Carlson's bunk with the intention of killing Lennie. The highpoint of the story comes when George has to kill his best friend, and the incident with the dog is mainly an excuse for revealing the existence of the Luger, where it is kept, how to fire it, and where to point it.
Whatever angle you choose to use in your essay, you need a specific thesis which you intend to prove. But you don't necessarily have to write the thesis statement first. You should be prepared to write several drafts.
Some people think it is important that Candy regrets that he didn't kill his dog himself. I think this would be a hard subject to handle. In the first place, Candy only has one hand. He couldn't hold the dog and fire the gun at the same time. He would be sure to botch the job. Besides that, there is an indication that he is right-handed and it is the right hand that is missing. So he would be trying to fire an exotic foreign handgun with his left hand!
Some people might try to work out an analogy between Carlson killing Candy's dog and George killing Lennie. I think this would be very complicated. Lennie is a friend, not a dog. If Candy killed his own dog there might be a better analogy, but I don't believe the author attached much importance to that parallelism.
My thesis would simply be: John Steinbeck created the incident in which Carlson shoots Candy's dog because the author wanted George to learn where the German Luger was kept, how it worked, and where to point it.