I have some questions with string theory:
Why is it that there is exactly 4 large spacetime dimensions while the rest remain small?
It is a nonlocal QFT. How could that fit in GR?
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The cosmologies with 6-7 small dimensions and 3+1 large ones evolving as in the usual 4D cosmology are solutions to the stringy equations of motion. They're not the only class of solutions, however. This really means that we can't derive the number of large dimensions from the first principles. Some people believe that the anthropic principle is the only answer – 3+1 dimensions is a very hospitable number for life.
A more old-fashioned scientific strategy was pioneered by Brandenberger's and Vafa's string gas cosmology,
See also its 500+ followups. These are ideas explaining the number by having a certain number of intersections of strings in Universes of various dimensions – and 3+1 is the maximum in which the large dimensions may keep on growing. These arguments are intriguing but remain inconclusive and it's even more true since the 1990s when the string gas cosmology had to be generalized to the brane gas cosmology which allows many different dimensions of branes etc. and everything is bit more complicated and ambiguous.
Concerning the second question, string theory is non-local in some sense because its fundamental objects are extended. So any reduction to point-like field theories will display some nonlocality. Also, string theory and any consistent theory of quantum gravity is able to get the information from the black hole interior to the exterior which requires some nonlocality, although very weak one, too. However, there are aspects in which string theory is perfectly local. One would have to go to very technical details to explain what I mean.
At any rate, general relativity is reproduced from string theory perfectly. The diffeomorphism symmetry holds exactly – in descriptions of string theory that contain the metric tensor including the unphysical polarizations as degrees of freedom (string theory adds infinitely many massive fields from the exciting strings, too, along with their own symmetries). String theory also perfectly satisfies the equivalence principle. At low energies, one also gets Einstein's equations. At shorter distances or higher energies, corrections start to become important but none of them contradicts any experimental test of GR or any general dynamical principle we associate with GR.
Moreover, GR with all of its glory is derived from a more fundamental starting point in string theory.