Was the original premise of U.S. involvement wrong? Primarily that policy makers interpreted a civil war as theater of a wider anti-communist struggle.

Whether or not the original premise of U.S. involvement was wrong is a question which will give you a myriad of answers from historians. In fact, whether or not the US was just or wrong in involving themselves in Vietnam is such a contentious issue in the study of the Vietnam War that one scholar dubbed it "Historians at War" due to the in-fighting and sometimes very harsh words being said about those on the other side of the argument.

There is also an issue of hindsight. Most laymen today would deem it to be a mistake to have gotten involved in the Vietnam War, but it is not always wise to see it out of the perspective of 2015. Instead, it's far better to consider the conflict from the perspective of 1955, 1960 and 1965.

Take the civil war interpretation for example. Popularized through the Academy Award winning documentary The Fog of War: Eleven Lessons from the Life of Robert S. McNamara, in which McNamara, who served as US Secretary of Defense, says that:

"In the case of Vietnam, we didn't know them well enough to empathize. And there was total misunderstanding as a result. They believed that we had simply replaced the French as a colonial power, and we were seeking to subject South and North Vietnam to our colonial interests, which was absolutely absurd. And we, we saw Vietnam as an element of the Cold War. Not what they saw it as: a civil war."

Now, this is his interpretation of what happened and shouldn't be completely discounted. However, that does not mean it is entirely right either. McNamara is making very broad claims about the belief of they (they who? North Vietnamese? South Vietnamese? The government of North Vietnam? etc.). The North Vietnamese government most certainly saw the US involvement in South Vietnam as a imperialist enterprise and young North Vietnamese conscripts were imbued with the message that they were going to "free your southern brothers from the chains of the imperialists".

The Vietnam War (between '65 and '73) definitely had aspects of a civil war, pitting neighbors against neighbors and parents having to see their children being conscripted into both the ARVN and the VC. However, insurgencies often involve these common aspects and the line between the two can be blurry. It is worth to remember at this moment that we are not speaking about a Vietnam, but two Vietnam. It wasn't the case of, for example, the American Civil War where a rebellious part of the nation rose up against the government and open warfare occured within months; instead, two separate nations came to be formed from the ashes of French Vietnam run and supported by nations with different ideologies. The chain of events that follow could be interpreted as a civil war, depending on your own approach and interpretation, but also of the invasion of one nation by another. This is also a very debated question amongst scholars. It would be incorrect however to say that they all saw it as a civil war when that simply isn't true.

An interesting note is LBJ's official goals and motivations for getting involved in South Vietnam. I've discussed it here previously.

How did military strategists approach the conlict and what were the principal aspects of its failure (or not as the case may be)

Regarding this, I have written a large post about it here. I am very happy to answer questions about this since this is actually my area of expertise.

What lessons were learnt by the military and policy makers?

While I will leave the question about policy makers to someone else, I will tackle that of the military.

Those that already had a tradition in counterinsurgency, such as the Special Forces, continued with their doctrines and specializations, but these were not brought on to the rest of the armed forces. They saw it as a really bad memory that had to be suppressed. The US Army genuinely saw conventional warfare as their primary purpose. It was, perhaps, their preferred purpose in this case. Just to show you how cynical the US Army was about counterinsurgency, when Colonel Harry Summers published his On Strategy: a Critical Analysis of the Vietnam War to the loud acclaimed from Army leadership, he genuinely stated that the US Army's path to defeat in Vietnam was because they were too focused on counterinsurgency. Yes, I kid you not. This only enhanced the view that conventional doctrine was the way to go and the conflicts that the US Military had to face after the Vietnam War didn't tell them anything else: Grenada, Panama and the Gulf War where all proof that the chosen conventional strategy was the primary purpose. Regrettably, the US Military was to be proven wrong on this as they entered the 21st century and the conflicts which had defined our modern age.