String field theory (in which string theory undergoes "second quantization") seems to reside in the backwaters of discussions of string theory. What does second quantization mean in the context of a theory that doesn't have Feynman diagrams with point-like vertices? What would a creation or annihilation operator do in the context of string field theory? Does string field theory provide a context in which (a) certain quantities are more easily calculated or (b) certain concepts become less opaque?
Dear Andrew, despite Moshe's expectations, I fully agree with him, but let me say it differently.
In QFT, we're talking about "first quantization" - this is not yet a quantum field theory but either a classical field theory or quantum mechanics for 1 particle. Those two have different interpretations - but a similar description. When it is "second-quantized", we arrive to QFT.
Feynman diagrams in QFT may be derived from "sums over histories" of quantum fields in spacetime; for example, the vertices come from the interaction terms in the Lagrangian, and the propagators arise from Wick contractions of quantum fields. This is the "second-quantized" interpretation of the Feynman diagrams.
There is also a first quantized interpretation. You may literally think that the propagators are amplitudes for an individual particle to get from $x$ to $y$, and the vertices allow you to split or merge particles. You may think in terms of particles instead of fields. In QFT, this is an awkward approach because most particles have spins and it's confusing to write a 1-particle Schrödinger equation for a relativistic spin-one photon, for example.
However, in string theory, spin is derived and the first-quantized interpretation is very natural. So the cylindrical world sheet describes the history of a closed string much like a world line describes the history of a particle. And it's enough to change the topology of the world sheet to get the interactions as well. So in string theory, one may produce the amplitudes "directly" from the first-quantized approach because the changed topology of the world sheet, which we sum over, knows all about multi-particle states and their interactions, too. We say that the interactions are already determined by the behavior of a single string.
Needless to say, like any Feynman diagrams, these sums over topologies are just perturbative in their reach.
Now, you may also write down string theory as a string field theory, in terms of quantized string fields in spacetime. Somewhat non-trivially, an appropriate interaction term - that "knows" about the merging and splitting of strings - may be constructed in terms of a "star-product" (a generalization of noncommutative geometry). In this way, string theory becomes formally equivalent to a quantum field theory with infinitely many fields in spacetime - for every possible internal vibration of the string, there is one string field in spacetime.
It used to be believed that this formalism would tell us much more than the perturbative expansions because, for example, lattice QCD in principle can be used to define the theory completely, beyond perturbative expansions. However, this belief has been showed largely untrue. At least so far.
It's been shown that string field theory indeed offers an equivalent way to calculate all the amplitudes of perturbative string theory - especially for bosonic strings with external open strings (closed strings are possible, and surely appear as internal resonances, but they are awkward to include directly as external states; superstrings are probably possible but require a substantially heavier formalism).
Also, string field theory has been very useful to explicitly verify various conjectures about the tachyon potential in bosonic string theory (or, equivalently, about the fate of unstable D-branes which emerge as classical solutions in string field theory). These investigations, started by Ashoke Sen, led to some nice mathematical identities that had to work - because string theory works in all legitimate descriptions - but that were still surprising from a mathematical viewpoint. But all the physical insights confirmed by string field theory had already been known from more direct calculations in string theory.
So because string field theory is widely believed not to tell us anything really new about physics, only a dozen of string theorists in the world dedicate most of their time to string field theory.
Moshe is surely no exception in thinking that it is not too important to work on SFT. Still, it is conceivable that sometime in the future, a more universal definition of string theory will be a refinement of string field theory we know today. However, it's also possible that this will never occur because it's not true: string field theory seems too tightly connected with a particular spacetime and with particular objects (strings) while we know that the true string theory finds it much easier to switch to another spacetime and other objects by dualities.
This is a matter of opinion, all I can state here is my opinion, and I’d fully expect other string theorists will have very different ones.
So, second quantization in field theory is a way to proceed from one particle QM to free field theory whose quantization will in turn give many particles of the type you started with. The point is that this is a way to introduce free (or by generalization, weakly interacting) field theory. It is a process which is perturbative in nature and does not provide us with non-perturbaive information. Nevertheless it is useful, for example it is easier to discuss things like the vacuum structure, off-shell observables, and some other issues which are completely obscured in a first quantized formulation.
Now, in string theory, the original formalism is first quantized, you discuss one string, or a fixed number of strings. It is natural to try to second quantize the theory, but in my view you'd be learning about perturbative string theory only in this process (as a sociological fact, all the non-perturbative knowledge we have about string theory does not connect very naturally to SFT). The formalism does have the advantage of having off-shell observables, like an ordinary quantum field theory, but it is not clear that in theory of gravity (which does not have local observables) this is necessarily a good thing.
This was all true for closed string field theory. There is a bit more motivation and results for open string field theory. Especially, there are some successes in exploring the space of (open string) vacua in this formalism (by discussing the so-called tachyon condensation and brane decay processes).
In this context I can elaborate a bit on what it means to second quantize the theory. You have creation operators that create an open string (in a particular state). You can add interactions (turns out you only need cubic ones) and create Feynman diagrams whose propagators and vertices are extended. Turns out that the set of Feynman diagrams you get is precisely the set of world-sheets you can build in open string theory. When you build loop amplitude, you can see that you have closed strings propagating as well, in intermediate states.
There is hope and some indications that quantization of open SFT will lead in the future to non-perturbative formulation of string theory (including the closed strings), but the state of the art is not very close to this goal, I think.
So, in my view the answer to both your questions is sadly negative, by and large, with some exceptions, but many really smart people continue working on the subject, and I may well be wrong in my expectations. I'd be happy if one of these smart people who has a more positive take will give their own answer here.
(I realize there may be obscure points in my reply, since I don't know your background. If you ask I'll try to explain things better).