That's a lot of questions! I'll have to diffuse my answer unfortunately.
As you said, words change meaning over time. The Latin was decimatio and had a military meaning, derived from the Latin for 'ten' (decim), effectively meaning 'to reduce by one tenth'. Despite its obvious Latin origin, 'decimation' as we use it today is a different word.
Flight was a natural and spectacular result of a unit's defeat by enemy warriors/soldiers and the will to fight of the defeated troops breaking down due to the imminent threat of death. Flight had terrible consequences for the defeated in the Roman era, routinely resulting in massive casualties for those fleeing as the victors pursued and killed anyone they could reach. Pursuits could last hours. An army also might not flee in its entirety, but units may flee yet an army might be victorious. It's a complex phenomenon but it was under no circumstances a good thing and an army could not be victorious unless it controlled the human impulse of its soldiers to flee when scared.
Yes it was effective, properly applied (ie. not too little not too much), coercion has always been an effective method of maintaining military discipline. It was used a lot in WWII, especially by the Germans and Soviets, and even American officers sometimes had to point pistols at their men to stop them from fleeing. Decimation and the various other punishments used by the Romans was a foundation of their superior military discipline, though decimation itself was used much less often than other punishments and individual executions.
It wasn't fair, it was terrifying. The purpose was to cow the remaining men into conducting themselves more bravely in the next battle for fear of suffering similar or worse punishments in the future. 'Fairness' had absolutely not a single thing to do with it, but it was an alternative to mass executions which, as the punishment for an individual fleeing from battle was technically death (fustuarium), would be impractical in the case of entire units fleeing.
Again, it's nothing to do with proportion or fairness. It's about winning battles, and the combination of Roman willingness to punish their soldiers in this way and the soldiers' willingness to accept such punishment even as citizens was essential to Roman success in battle.
Again, it had nothing to do with fairness. A Roman maniple or cohort was expected to fight as a unit, and the shame of the unit was shared by every individual. For instance, this applied to the unit losing their standard. Obviously only the men at the front lost the fight, but it was generally the guys at the back that fled first, and nonetheless the entire unit was held responsible.
Manpower was not something the Romans were short of. Numbers were a lot less important than the combat motivation of the soldiers and the morale of the army.
The general would oversee the entire process, the tribunes would manage it more closely, and the soldier chosen to die was beaten to death by the other nine men in his group of ten.
It was death by beating with cudgels, just as per fustuarium. It was public, it was painful, it was shameful, and this was the point: to terrify the witnesses into doing better next time. I'm not sure what they did with the bodies, they were likely buried. They may have been left displayed or piled somewhere they could be seen as a reminder of what had occured for a short time.
As other users have linked, there are numerous examples of decimatio being used in the Republic. It was not always at use during the Republic, it may have only been used for periods of the 2nd century and 1st century BC, and it appears to have mostly fallen out of use in the professional Imperial army. The decimation by Crassus in the Third Servile War is the most well-known example.
I should emphasise that it was probably used quite rarely, but it was certainly in use during the mid-2nd century BC, hence Polybius' detailed description of the practice (Pol 6.38). As you have observed, there was a fine balancing act to be found between motivating men to try harder next time, and shattering their confidence. Shaming punishments were generally more effective, and are far more common under commanders we know a great deal about such as Caesar, and were more common in general in the Empire. For more information you should read: Goldberg, Charles. “Decimation in the Roman Republic.” The Classical Journal, Vol. 111, No.2, 2015, 141-64. You can at least read the abstract and the first page of it if you have no subscription here.
Treating cowardice as a capital offense has a long history, and it's still a capital offense in the US military. The last US serviceman to be executed solely for cowardice before the enemy was Pvt. Eddie Slovak in World War II. Knowing that his unit was expecting to face combat, and having been terrified during previous combat, he simply walked away from his unit and turned himself in with a note admitting to his action. While he expected a prison sentence, he instead got the death penalty, which was carried out by firing squad, the only case of execution for cowardice carried out by the US in that war. (Other executions occurred, but were for other crimes like rape and murder, and other cowardice death sentences were commuted.)
The reason for this sentence has to do with the men left to do battle. At best, a deserting soldier had to be replaced by someone else who may die in combat, a sort of indirect form of murder. Worse, the unit may be left short-handed, and others may die because one man sometimes does make all the difference. At the absolute worst, it can inspire others to desert, costing even more lives and maybe even the entire war. If one's choice is maybe die in battle or certainly die by execution, perhaps one is more likely to stay with one's unit.
Just FYI, if you want to check an etymology, the Online Etymology Dictionary is a good place. If you just wanted to ask about decimation, then carry on.