I agree that executions of those who have committed murder is moral. And the reasons to hesitate to use execution are mostly about the degree of certainty we can obtain in the convictions. But I disagree with your approach to justifying executions.
Murderers are not humans.
I disagree with this statement. And I don't believe it is Objectivism's approach to this issue. The reason it can be moral to execute a person is that they have lost their right to their life when they become a murder.
What characterizes them [murders] is their hate against human beings, which is in itself sufficient proof that they are not humans; for feelings, as philosopher Ayn Rand taught us, are not tools to obtain knowledge.
Psychologically, there are many people with very low self-esteem who project their self-hatred onto others - even onto the very idea of humans. There are countless people who hate against human beings but never commit murder. To "hate against human beings" is not in itself sufficient (or necessary) proof that they are not humans.
It is true that feelings (emotions) are not tools of cognition, but they are evidence of inner workings in our minds, and they are positive experiences to be sought or negative ones to be avoided and we should nearly always pay attention to our feelings and emotions rather than to be blind to them. And they are part of being human (a very good part when one follows the practices resulting in good self-esteem).
The foundation and the means to obtain the human condition, is the faculty of reason. Hence, to state that murderers are not human beings in the proper sense of the term is the logical consequence of Ayn Rand’s teachings.
Actually, people can be irrational and not be murders. People can be irrational and still be human. There is a confusion here on the use the word "reason" in the phrase "faculty of reason." Man's faculty of reason is a faculty whereby individuals exercise their choice to be reasonable or to be unreasonable in a given instance. That is, a person choosing to be irrational, or unreasonable, is exercising their faculty of reason - just in the wrong way. Rand says, "Man’s essential characteristic is his rational faculty," and that it must be exercised by choice. And that there is no guarantee that a man will get all things right.
A murder is a human being that has violated another's right to life. No one can violate another's right to life without stepping totally away from their own right to their own life. That is why it is not immoral to execute a murder - because they have volutarily relinquised the right to their life when they committed a murder.
People are responsible for not violating others' rights - someone might kill in burst of drunken rage, but they are the ones who chose to get drunk and they are ones who chose to act on that rage and they are one who violated someones right to life. The drunken rage can be an explanation, but never a moral justification.
Their brain may have developed up to the level where the faculty of reason starts to operate, but they rejected it before it reached its full extent or, else, they alienated the minimum of reason that their brain was able to retain from its original function of providing the correct sustenance for a life proper to a human being.
Nothing in neurophysiology sustains that assertion that a murder necessarily has a different brain, or that the difference in the brain was due to rejecting reason or that they now have the brain of a non-human animal. Is it possible that you meant to use the word "mind" and not "brain"?
The term of murderer is not limited to the physical act of the murder itself, i.e. the actual execution of a maliciously premeditated killing, but refers also to hate mongers, religious, political and military propagandists who seek to promote hatred and prejudice against another group or those who do not follow their same beliefs, wishes and aims.
That completely changes the definition of murder. And it totally rejects Ayn Rand's view of individual rights in which it required an act of initiating physical force against another, threats of physical force against another, fraud or theft to violate a right. You are now advocating a kind of censorship where you are making some ideas or speech criminal offenses and that isn't even near Objectivism. Rand was quite clear that an act was required to violate the rights of another and that the only acts capable of doing that were those that took away choice and that those only included intiation of force, fraud, or theft fit that bill.