1. client's blood. Killing the Captain would, therefore, sully his reputation and eliminate his respectability.
2. The barber believes that murder is wrong because it goes against his principles. He identifies as a "revolutionary" and this is, in his opinion, very different from being a murderer.
3. The barber realises that murdering the Captain would create more problems than it would solve. Instead of ending the conflict, for instance, it would enable it to continue because members from both sides would use murder to bring about retribution: 

"Others come along and still others, and the first ones kill the second ones and they the next ones and it goes on like this until everything is a sea of blood."

Above all else, the barber did not want to commit the greatest sin of his profession--to spill a drop of the customer's blood. Killing Captain Torres while performing his barbering duties went against the unwritteen barber's code. Additionally, the barber considered it a cowardly act of murder, killing the captain while his eyes were closed. He believed that killing the captain in this manner would make him no better than his victim. Also, the barber knew that he would be identified as the killer--murderer or hero, he knew not which--and he had nowhere to go to hide from the men who would be hunting for him. Killing the captain would also reveal his true identity as a spy for the rebels, and by blowing his cover, he would no longer be useful as an operative.

The reasons that the barber gives in favor of killing Captain Torres have to do with Torres’s actions. Torres is a bad person who has done a lot of harm to a lot of people. The barber is an informant to the rebels who are against everything that the captain does and everything that he stands for. The barber thinks about how if he killed him, he would be helping his own cause and protecting the captain’s future victims. Consider what the barber thinks to himself while he is getting rid of the captain’s beard:

How many had he ordered shot? How many of us had he ordered mutilated?

These questions show the reader that the reasons the barber lists for killing the captain have to do with the numerous violent crimes the captain is responsible for. Once the barber is done shaving the beard, he goes on to think,

He is a calm man, who is not even thinking about what he is going to do this afternoon.

Here the barber is reflecting on the captain’s character. He is noting Torres's indifference and insensitivity to the violent things he does, which supports the option to kill him.

The barber also thinks about how easy it would be for him to kill. He notes that if he just pressed a little bit harder on the razor, then he could easily pull it off. But in the end, the reasons he lists against killing him are more convincing.

In this story, the Barber is secretly working as an informant for the rebels. Captain Torres, on the other hand, is tasked with bringing down the rebels and has already killed a great number of them. This explains why the Barber wants to kill him: it is an act of revenge for the murders that the Captain and his men have committed. It will also prevent the Captain from killing any more.

Later in the story, we learn of another reason why the Barber might want to kill Captain Torres, and it is expressed through the following lines:

And then on the other side. "The avenger of us all. A name to remember. (And here they would mention my name.) He was the town barber. No one knew he was defending our cause."

In other words, the Barber knows that if he kills Captain Torres, he will become a hero among the rebels. His prestige and reputation will be massively enhanced because he has killed the rebels' number one enemy.

Despite these motivating factors, the Barber is unable to kill the Captain. It seems that his desire to be a peaceful man and to simply be a barber outweighs his desire for revenge and prestige.

Perhaps the biggest confict in Hernando Tellez's short story, "Just Lather, That's All," is the barber's indecision about how to use his razor while he has Captain Torrez in his chair. As a barber, his primary function when shaving a customer is to deliver a clean cut without spilling a drop of blood. However, the barber has a secret and a dilemma:

... Yes, I was secretly a rebel, but I was also a conscientious barber, and proud of the preciseness of my profession.

The barber wants to kill Torrez because the Captain is the primary rebel foe: He hunts them down, kills them, tortures them, mutilates them. Killing Torrez would avenge the death of so many of his cohorts, but someone else would only replace Torrez and continue the killing. So, the barber decides to maintain his rebel anonymity and let his adversary live.

The barber eventually comes to the conclusion that he will not kill Torres because he discovers something very central about himself, his identity, and what he can and can't do. This epiphany also recognises something central about the identity of Captain Torres. Note what the barber concludes as he makes his decision not to kill Torres, in spite of what he had done and what he will continue to do:

But I don't want to be a murderer, no sir. You came to me for a shave. And I perform my work honorably... I don't want blood on my hands. Just lather, that's all. You are an executioner and I am only a barber. Each person has his own place in the scheme of things. That's right. His own place.

The barber recognises that, to him, what is most important is honour. As a barber, it is vital to him to do the best he can in his work and job, which means giving the best shave he can. He also recognises that "each person has his own place in the scheme of things." Even though the place of Torres is clearly to kill and murder, that is his place, just as it is not the barber's place to kill. The identity of Torres is defined by his being "an executioner," whereas the barber's identity is defined by the job that he does. The barber therefore decides not to kill Torres because he realises that this is not part of his central identity.

When the Captain first approaches, the barber trembles with fear and, while shaving the Captain, most of his meditations are similarly fearful. However, immediately before he decides not to kill the Captain, his thoughts take a moral turn and he decides he doesn't want blood on his hands, only lather. He lets the Captain go free because, despite the brutal surroundings and situation, in the midst of civil war, he will not be such a brute as to kill another human being. As well as being brutal, such a murder would arguably be peculiarly cowardly, as a man in a barber's chair is in the most helpless position anyone can imagine.

The barber would therefore have been cowardly twice over if he had killed the Captain: for going against his moral principles and for slitting the throat of a defenceless man. He would also, presumably, have been hailed as a hero by the revolutionaries. Cowardice and heroism are not mutually exclusive.

As the story stands, it is at least arguable that the barber is courageous in sticking to his principles but refuses to be a hero, since he does no deed of daring. He is, therefore, neither a coward nor a hero.

To answer this question, it is important to consider it from different perspectives. From the perspective of the rebels, for example, killing Captain Torres would make the barber a hero. This is because the captain has killed many of their comrades and, as a result, they view him as their enemy. At one point in the story, the barber alludes to this heroic status when he is considering killing the captain:

"And then on the other side. "The avenger of us all. A name to remember. (And here they would mention my name.) He was the town barber. No one knew he was defending our cause."

In contrast, some people, like the captain's men, might call the barber a coward for killing Captain Torres. After all, only a coward would kill an unarmed man while giving him a shave. At the end of the story, the captain appears to judge the barber as a coward when he reveals that he knew he might be killed:

"They told me that you'd kill me. I came to find out."

So, in an ironic twist, the captain views the barber as a coward, even though he has, in fact, saved his life. 

The decision that the barber has to make will either result in him being regarded as a coward by the rebels and a hero by the military, or a coward by the military and a hero by the rebels. Consider how he balances the way that his actions will impact his future and how others will think of him:

"Captain Torres' murderer. He slit his throat while he was shaving him--a coward."

And then on the other side. "The avenger of us all. A name to remember. He was the town barber. No one knew he was defending our cause."

Therefore we can see that the two opposite reactions depended entirely on the people that you are talking about. The rebels would have considered the barber a hero, whereas the military would have hunted him down like a dog and killed him for a coward. The barber finds himself in an immensely difficult situation, where acting or not acting is going to have massive consequences on the rest of his life.

To answer this question, take a look at what the barber says on the subject of killing Captain Torres:

I could cut this throat just so, zip! zip! I wouldn't give him time to complain and since he has his eyes closed he wouldn't see the glistening knife blade or my glistening eyes.

In other words, Captain Torres has no idea that the barber is helping the rebels. In fact, he thinks that the barber is just a barber, so he has no reason to expect that he would be murdered while having a shave. As a result, Captain Torres would not have the opportunity to defend himself against the barber's razor. He would have no warning of the violence about to occur. Moreover, after killing the captain, the barber would be forced to flee and hide because the captain's men are sure to seek revenge.

For the barber, then, murdering Captain Torres would be a cowardly act. This idea contributes to his decision not to murder the captain. In addition, the barber is proud of his profession, abhorred by the notion of murder, and simply wants to be seen as a "revolutionary," not a criminal.

Killing a man by slitting his throat while he sits before you with eyes closed seems to be a pretty cowardly act to me. Accepting money for a service and then committing the act only makes it more shameful. When Captain Torres came into the barber shop for a shave, he probably knew that there was a possibility that he would not leave alive. But he must have judged the barber correctly: If indeed he was one of the rebels, the public murder would not only blow the barber's cover--thereby rendering him useless for future information--but also possibly turn public opinion against the rebels for committing such a murder. Few acts of murder can be called anything but cowardly, but the method that the barber would have been forced to take--cutting the throat of a defenseless man from behind--would be hard to defend.

Whether the barber should have killed Captain Torres is a question that heavily depends upon one's personal feelings on revenge.  Some might argue the barber had a duty to kill the Captain and avenge those poor rebels he has killed or ordered killed.  The reality of the situation is much more complex.  Captain Torres is obviously not the only person involved in the mission of tracking down the rebels.  Therefore, it is reasonable to assume there is a Lieutenant or other subordinate at the ready to assume command of the mission.  The killing of Captain Torres may bring about even more scrutiny on the tiny village.  The barber must also think of these things. 

The other matter the barber must address is his own conscience.  He states quite clearly he is not a murderer and trembles when the Captain walks into his shop.  The barber has his own pride about his job.  The barber had much more to lose than gain by killing Torres.  The rebels may have come under greater threat and his own conscience would suffer.  In the end, there was little choice for the barber but to cut Captain Torres' beard and let him walk out of his shop.

Your original question infringed enotes policy by asking more than one question and so I have had to edit your question down to just one question. This is the kind of question to be honest that you can answer either way - as long as you support your response fully. The text gives us no clear indication so you need to think about your own personal response to the barber's decision to not kill Captain Torres and think about whether it is correct or not.

Personally, I think at the end of the story the barber experiences a kind of epiphany that means that he has a moment of realisation about himself and his place in the world. He realises who he is and what he is and is not capable of:

But I don't want to be a murderer. You came to me for a shave. And I perform my work honourably... I don't want blood on my hands. Just lather, that's all. You are an executioner and I am only a barber. Each person has his own place in the scheme of things.

Thus the barber realises something fundamental about his identity - he is a barber, not a murderer, and therefore he makes the right decision by being true to who he really is and not killing Torres. In return, Torres himself comes to respect the barber for who he is, just as it seems the barber respects Torres and the role that he plays.

Captain Torres obviously believed that the barber must have been associated with the rebels, and he was willing to do absolutely anything--including putting his own life in danger--in order to discover whether the barber was a spy. Torres deliberately revealed that he had captured another group of rebels and that he would be torturing and killing them later that day. Torres hoped that the barber would contact other rebels so he could round them up as well. The captain also expected that the barber would do (or at least attempt to do) as he had suspected--cut his throat while being shaved. By doing so, Torres would expose the spy, even at the expense of his life.