The short answer (these are admittedly all short and partial answers) to the first question is simply that every nation and organization needs a person or a body of people that can settle issues once and for all, a place where, as Harry Truman put it, the buck stops. The alternative is chaos often followed by violence. While Americans want nothing to do with kings or emperors, we, too, need a source of final authority. Think of the mayhem that might well have occurred following the election of 2000 without a Supreme Court to halt the shenanigans going on in Florida. The Court continues to play this role because it works, and we are a practical people. We've had our Civil War and are now content (at least the vast majority of us) to work together as a single nation under law, as defined by the Supreme Court.
The problem with this answer is that it is straightforwardly false: the Supreme Court does not, and by its very nature cannot, "settle issues once and for all." Since the Supreme Court works by establishing precedent rather than by imposing decrees, and since precedent admits of continual change, reinterpretation, modification, qualification, and reversal, no matter of general principle is settled by the Supreme Court 'once and for all'; all decisions on this point are temporary and admit of potential qualification within a very short time. Moreover, since the application of precedent is always by analogy and parity rather than by any sort of direct application, every Supreme Court decision by the nature of the Supreme Court leaves open and even raises as many questions as it answers. This, in fact, is why Congress leaves so much to the Court in the first place: Congress can leave some unpleasant responsibilities to the Court without having to worry that Congress itself will become otiose in the process.
Moreover, the answer overlooks what I would have taken to be the usual point when people ask this question, namely, that if you are going to have anyone with the responsibility to "settle issues once and for all" and if you are going to establish anything as "a source of final authority," it should be more like an elected Congress than like an appointed Court, or at least should be an elected Court. One can certainly imagine any number of alternatives to the system we have; one could have, to take just a fairly extreme example, a system in which the role of the Supreme Court was taken by a Judicial Congress whose members were elected by the people and confirmed by the legislature, which issued not precedents but directives. The many problems of such alternatives would have to be addressed, but it really doesn't answer the question to say, "Oh, well, we need someone to do it."