Are quantum mechanics and general relativity predictions of string theory, or were they built into the theory from the start? If the former, why aren't tests of GR and QM considered simultaneous tests of string theory?
To address your last question:
why aren't tests of GR and QM considered simultaneous tests of string theory?
Physics progresses by devising mathematical models with ever widening scope. For example Newtonian gravity is a subset of General Relativity that applies when densities and velocities are not too high. So it is a test of General Relativity that it must reproduce the predicitions of Newtonian gravity in this limit.
So testing Newtonian gravity could have shown GR was wrong. However because there are possible alternative gravitational theories that also reproduce Newtonian gravity, testing Newtonian gravity cannot prove GR is correct.
Likewise, string theory could have been proved wrong if it did not reproduce the predictions of GR and Quantum Field Theory at low energy, but the fact it does reproduce GR and Quantum Field Theory is not proof that it is correct. There could be other theories that reproduce GR and Quantum Field Theory in an appropriate limit.
Answer to your first question - The principles of quantum mechanics is built into string theory from the start. However, general relativity is a consequence/prediction of string theory.
To answer your next question, note that QM and GR have different non-overlapping regimes of validity. Within their own regime, each is a "correct" theory. The new thing that string theory does is to extrapolate either theory outside their domain of validity and thereby allow for study of quantum mechanics on the scale of galaxies (however small these effects might be) and similarly the study of GR on the scale of atoms. In order to truly verify string theory, one needs to study the theories in these extended regions. Simply studying QM for atoms and GR for galaxies does not validate string theory, it only serves to validate QM and GR itself.
So, the next question is - what effects can one look for to validate string theory. To answer this, let us be more precise about the domains of validity. Quantum Mechanics is valid in "small regions". Let us roughly denote the size of this volume as $V \sim \epsilon^3$ for some small $\epsilon$. Now, in GR a length scale is set by the radius of curvature of the spacetime $R$. In particular, if we are at length scales $\ell \ll R$, then all effects of gravity may be neglected. Thus, in order to test both QM and GR simultaneously, we must have $R \sim \epsilon$, i.e. we must work in a background that has an extremely small radius of curvature.
Such backgrounds are available in the vicinity of black holes. Thus, black holes provide the perfect arena to test string theory. However, there is another problem - There are two levels at which one can discuss a quantum theory in the presence of gravity -
QFT in curved spacetimes - Here, one quantizes everything except gravity. In this case, a lot of interesting phenomena such as BH thermodynamics, Unruh effect, Hawking radiation, etc. arise. While these phenomena can be studied near black holes, they do not provide any sort of validation for string theory itself.
Quantizing gravity - This is where string theory truly comes in. It quantizes not only particles, but also gravity itself. Such quantization, string theory predicts gives rise to lots of exotic phenomena such as compact dimensions, tower of particles, etc. These are really they things that one needs to probe in order to have a true verification of string theory.
As you can imagine, doing the latter is the hardest of the lot.
PS - In principle, I have been lying to you a little bit. While it is true that real tests of string theory can only be done in the extreme conditions described above, there are experiments that we can do here on Earth to validate it.
For instance, string theory requires supersymmetry. Without it, it fails (atleast as it is formulated today). One can test for the existence of supersymmetry at, say, the LHC. If we find it, good. But if we don't, that is a serious problem for string theory.
There are many more similar experiments that we can conduct. I'll leave to peruse this answer by @Lubos which describes these really well.
String theory is a quantum theory. Quantum theory is not a string theory. General relativity is a low energy limit of string theory.
String theory is a theory that put together the equivalence principle and quantum principles.