Is there a minimal string length (maybe the Planck length), and is it quantized?
Do strings have a 0-dimensional (ie point) cross-section?
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This is a very interesting question. It is said that string theory is free from adjustable parameters, but the length scale for string physics is set at the Planck length. Being a fundamental theory of forces, including quantum gravity of course, is natural to set the string scale length to the Planck length because this is the distance where quantum gravity effects should become important. Since the actual quantum gravity theory is not known, to set the string length to the Planck’s length is like an educated guess. As string theory advanced during the years, the possibility of the Planck’s quantities being dependant on the geometry of space-time make necessary to question what is the actual Planck length in our 3-D brane. Since a real string’s length has not being actually measured, no one can be certain that it’s the Planck length, although this is the most probable number if strings actually exist at all. With regards to your second question, in string theory, as is actually formulated, is based on a fundamental string of length lp. I don’t recall a string length quantization principle other than the winding mode quantization for closed strings around acompact dimension. It is assumed that the minimum string length is fixed by the interplay of string tension causing it to contract and the uncertainty principle causing it stretch. For the moment this number is a parameter and not a dynamical quantized variable. If an open string is attached to different branes it is assumed that it can be somehow stretched giving the brane system more tension (energy) but this doesn’t actually explains why the minimum string’s ht length is the Planck length. Also the value in G is actually set by the dilaton field vacuum expectation value, making the argument a little bit circular. As I said before this is a very interesting question that goes to the heart of what string theory is.
This is one of the most confusing points when learning string theory. The theory is superficially formulated on an unquantized flat background, and it looks like there is nothing quantum about the space. The string length is not quantized in the way you mean (there are string excitations of different classical length, but the space-time itself is continuous). Still, the quantum string reproduces GR at long wavelengths, and it has a consistent expansion, but it is unnerving that it doesn't look like quantum gravity and the expansion is precisely in a limit where the string doesn't probe the interesting quantum gravity stuff you expect at the Planck scale.
The reason is that the perturbation series for string theory is at small string coupling, so that the strings do not interact at all to lowest order. In this approximation, the gravitational coupling is very weak, and the value of "G" that you would measure by scattering strings is very small. This is taking the weak coupling limit while holding the string length fixed. If you take the weak coupling limit holding the Planck length fixed, what this means is that weakly interacting strings are very long compared to the Planck length, and are big floppy things that are averages over many Planck lengths of whatever is going on down below.
This is the reason that you can formulate string theory at all consistently--- there is a weak coupling scaling limit where the strings are weakly interacting and enormous compared to the Planck length, and the problem of quantizing the gravity at small scales is sidestepped by quantizing the string motion instead. Once you do this, you find that you have described all the interactions of the objects in the theory, including gravitons, to leading order in string coupling, so you conclude that you have a perturbative gravity theory.
But this doesn't mean that you have a consistent description of what is going on at short distances, since you never dealt with the structure of space time except through external fields in the string action. Suppose you try to fix this, by scattering weakly interacting strings at energies high enough to probe the Planck scale. Then you need to deal with very high orders of string perturbation theory, and the perturbation theory is going to break down. This property made me think that string theory is nonsense for a long time, because it seems like cheating--- you are formulating a gravity theory without dealing with any of the problems of quantum gravity!
But string theory is correct despite this initial misgiving. String theory is very sophisticated philosophically, it only defines the geometry to the extent that you can see the geometry by scattering things from infinity. This is the S-matrix point of view. The modern way to say the S-matrix idea is through the holographic principle--- the gravity theory is only defined on the boundary, and the microscopic geometry is reconstructed from a consistent scattering in the interior.
Using weak coupled string probes, you sniff out a long-distance approximate geometry, but if you want to probe short distances, you need other probes, and these are provided by the D-branes. he dualities of string theory demonstrate that the strings and the branes are not two different things, but string theory happens in a limit where the branes become long and light, and have a consistent peturbation series only among themselves, because all the branes become heavy. It is fortunate that these pure string limits exist, otherwise we might never have discovered the theory.
But in the pure string limits, the answer is no. The space-time is not quantum in any way, the strings are propagating on scales where the geometry is classical, and you only see the interesting quantum stuff using string dualities and modern formulations, like AdS/CFT or Matrix theory. The 1990s part of string theory is essential to making it a complete quantum gravity theory, 1960s-1980s string theory was just the first step.