[Copernicus]

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1. No
Nobody has ever thought about the punishment regarding certain crimes before committing them. Everyone thinks they won't get caught therefore punishment for crimes is completely meaningless as a deterrence. (especially so from a life vs death sentence difference--is there really one in a criminal's mind? I doubt it)

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On the issue of deterrent value, you cannot cite people who have committed crimes in order to prove that the deterrent value is minimal, as those people were undeterred. There will always be the undeterred.
From that point, the question isn't really "does the death penalty deter" as much as it becomes "are we executing capital punishment in such a way that it serves in its proper deterrent capacity"

Like I said earlier... This question asks to prove an unprovable negative, as do all questions that query about the value of preventive measures.
Prove that if I had not changed the oil in my car, it wouldn't have broken down for 100,000 miles...

As an aside, I do believe that the death penalty is of immeasurable value when it is open and made public for all to see. When it is behind closed doors and done in secret, then it becomes nothing more than a vengeful, spiteful action undertaken by the state.

Another question: Say someone murdered the family member- the mother- of another person. The murderer was convicted, and summarily sentenced to death.

On the day of his execution, while being led from the holding cell to the injection chamber- mere minutes before he is about to be executed- the son of the murdered mother shoots the murderer with a rifle from the field across from the prison yard and kills him a couple feet from the entrance to the executioners pen.

Is the son guilty of murder? If so, then how is the state justified to do the same thing mere minutes later? If not, then what is the value of having the state complete the executions in the first place, if their main value is for the benefit of the families?

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Change your scenario slightly to this: The law allows a family member of the victim to drop the cyanide tablet/pull the switch/inject the serum.

What is the difference between the two scenarios?

Society as a whole has determined that the death penalty is appropriate, and it is society's responsibility to carry out its wishes.

In your initial example it is still murder (barring insanity/extreme distress and similar defenses), because society has installed itself as a proxy to carry out the family's revenge andthe family member has taken society's role upon himself.

In my changed example he is conforming to society's rules.

(BTW I am pro death penalty and but would not be in favor of allowing a family member to carry out the execution. Distancing the family member from the act itself helps to insulate him from feelings of remorse for the execution, which I believe has more value than the act of execution itself.