In the sense of "Copenhagen Interpretation", what exactly is an interpretation? What purpose does an interpretation serve?
I would describe interpretations of quantum mechanics as part of the philosophy of physics. Here is a well-known quote by Bertrand Russell: "As soon as definite knowledge concerning any subject becomes possible, this subject ceases to be called philosophy, and becomes a separate science." This mathoverflow answer and the comments on it may also be helpful. The Copenhagen interpretation (CI) was the first interpretation of quantum mechanics, and it arose at a time when there was honest confusion and doubt about what quantum mechanics meant. The "shut up and calculate" option was not necessarily available, because the rules for calculation weren't yet clear. The CI was developed around 1925-1927. Here are a couple of examples of developments that occurred during that period, which show how unclear the situation was. The Heisenberg uncertainty principle dates to 1927 (and was not yet properly understood when Heisenberg first published it). Also, Bohr had been pushing the idea that quantum mechanics applied to matter but not to light (the Bohr-Kramers-Slater theory), and this was disproved experimentally by Bothe and Geiger in 1925. So the general idea is that this was a period when people were struggling to achieve the kind of "definite knowledge" referred to by Hilbert. After 1927, they were well on the road to "definite knowledge," and the philosophy of quantum mechanics had made itself nearly obsolete -- which, in in Hilbert's description, should be the goal of every sub-field of philosophy. One way we can tell that this process of self-obsolescing was completed after not much longer is that in the postwar period, American textbooks almost all omitted any discussion of interpretations.[Osnaghi 2009]
Can an interpretation be tested or even be correct or incorrect independent of the underlying theory being interpreted?
The part about "independent of the underlying theory being interpreted" is important. It restricts us to talking about testing one interpretation against another, as opposed to testing quantum mechanics in general. The Copenhagen interpretation's main competition is the many-worlds interpretation (MWI), which came much later, in 1957. WP has a bunch of material on this topic, which I think gives a fair picture of the scientific consensus.
MWI is considered by some to be unfalsifiable and hence unscientific because the multiple parallel universes are non-communicating, in the sense that no information can be passed between them. Others claim MWI is directly testable. Everett regarded MWI as falsifiable since any test that falsifies conventional quantum theory would also falsify MWI.
(The second reference is to [Everett 1957].)
If you look through this whole section of the WP article, I think you'll see that there's an overwhelming consensus that interpretations are not testable in the sense of testing one interpretation versus another. As suggested by the quote above, they are only testable in the sense that one could try to falsify quantum mechanics in general.
I think MWI was a positive contribution because before it came along, there was only CI, and one could have gotten the impression that CI was the only possible correct interpretation of quantum mechanics. (Many people still have this impression.) MWI was a good antidote to this kind of smugness. It showed that when we talk about what quantum mechanics means, there is the potential for a huge amount of ambiguity, and this ambiguity is never going to be settled. That is, according to our current understanding of the standard model, there is no experiment, even in principle, that could ever decide between CI and MWI. Therefore we can tell that any question that is answered differently by CI and MWI is a question that can never be answered empirically. For instance, one can never determine empirically whether quantum processes are "really" random as in CI, or whether the evolution of physical states "really" goes through non-unitary steps. A discussion of the complicated historical context is given in [Osnaghi 2009].
Another salutary effect of MWI was that it raised the issue of what "measurement" was in quantum mechanics. For example, Stern, describing Bohr's view of Everett's work, said, "[...] the basic shortcoming in his method of
approach [...] is his lack of an adequate understanding of the
measuring process." This has to some extent been clarified in terms of decoherence.[Zurek 2001]
Everett, "Relative State Formulation of Quantum Mechanics," Reviews of Modern Physics 29 (1957) 454–462. Everett's thesis on this topic is available online: http://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf
Osnaghi, Freitas, and Freire, "The Origin of the Everettian Heresy," Studies in History and Philosophy of Modern Physics 40 (2009) 97–123, http://stefano.osnaghi.free.fr/Everett.pdf
Zurek, "Decoherence, einselection, and the quantum origins of the classical," 2001, http://arxiv.org/abs/quant-ph/0105127