How does String Theory predict Gravity? Firstly, General Relativity states that Spacetime is dynamic and is consonant with the distribution of matter/energy. How does String Theory predict gravity, when it is background dependent, that is it starts off with a fixed, and presumably flat space-time?
Secondly, String Theory predicts the existence of a graviton, but seeing we don't have a quantum theory of gravity, how do we even know that this is the correct quantization of gravity? In other words how do we verify that the graviton should have the properties that String theory says it does, that is what grounds do we have for believing in the properties of the graviton?
Thirdly, how do we go from the existence of gravitons to the curvature of space-time? When we talk about gravitons aren't they already existing in some space-time framework?
 A: In general relativity the gravitational field is given by the metric tensor in space-time. The metric tensor is the solution to Einstein’s field equations. It is a symmetric tensor with ten degrees of freedom. So fundamental excitation in quantum gravity must have spin two. In string theory the oscillations of the closed string includes a symmetric tensor state that can be identified with gravity as well as anti-symmetric tensor (The Kalb Ramond field) and a scalar field (the dilaton). To get Einstein’s GR field equations you have to find an effective low energy action that is fully compatible with the quantum conformal symmetry of the original closed string theory describing the massless bosonic excitations at the classical level. So the GR field equations are just the field equations obtained from the low energy massless bosonic action of closed string theory. This way you can even generate corrections to GR by going to higher level terms in the double curvature and string interaction expansion. This process of obtaining effective gravity equations is well described at an easily understandable level in Maurizio Gasperini’s String Cosmology book. In that book is also explained how to describe strings in general curved backgrounds.
About you third point. gravitons are usually described in the context of linear GR. In this description the metric tensor of space-time is separated in two parts the background metric and a fluctuating part that is then quantized. So you can visualize a graviton as a quantized oscillation describing the fluctuation of space time with respect to the background metric. So you can not say that the graviton is propagating in space time, the graviton is a quantized fluctuation of the space-time itself, not just propagating in it.
In the context of string theory you can visualize the graviton as closed string states present in regions of fluctuating curvature of space-time. This is analogous to thinking about photons and electromagnetic waves. The EM description is valid when the number of photons is high. You can say that a region in the wave with large amplitude is rich in photon states. Gravitons closed string states and curvature can be seem the same way. A region of space where curvature is high is rich in closed string states that are a part of spacetime itself. This is a simplified and rough version of what gravity waves are in string theory.
A: Someone -- Richard Feynman apparently [nope, I was wrong! see comment by @RonMaimon below] -- decided at some point to see if the massive success of QED theory could also be applied to gravity. Attempting to represent gravity as a force mediated by bosons (force particles) produced spin-2 carriers for the gravitational force, what we now call gravitons. When loops in early versions of string theory predicted similarly spin-2 particles, it was (and is) assumed that the two concepts are basically the same.
Alas, if any truly elegant reconciliation exists between Einstein's decidedly non-quantum curved-space model of gravity and the attempts to make gravity a matter of particle exchanges, I've never heard of it. I think the quantum model of gravity sort of won by default, perhaps in part because it provided a new and interesting set of mathematical issues to pursue.
