Currently, the dynamical principle which governs the form of compactified dimensions is not known. It seems that one has to understand String Theory non-perturbatively in order to be able to find the form of entire 10D spacetime doing some pre-Big Bang string cosmology (see, say, this book by Baumann and McAllister on String Inflation where the String Theory implications to inflationary physics are presented) dynamically from certain symmetric initial condition. There are some phenomenologically reasonable stationary compactifications of String Theory which lead to low-energy 4D physics resembling the Standard Model. See, for example, the 2nd volume of Green, Schwarz, Witten.
The role of AdS/CFT is two-fold. First, it allows one to describe full String Theory on a certain background, which is, in particular, a quantum theory of gravity, using techniques of 4D SUSY gauge theory, which is much simpler. The problem is that even 4D gauge theory is still too complicated to be studied quantitatively in most interesting range of coupling constants. The most famous example of ${\cal N}=4$ $SU(N)$ SYM theory dual to Type IIB String Theory on $AdS_5 \times S^5$ background with $N$ units of 4-form cuvature strength through $S^5$ allows one to study String Theory using SYM only for the case of $g_{YM}^2 \ll 1$ and $g_{YM}^2 N \ll 1$, when the perturbative QFT works well. Second, it allows to study strongly coupled gauge theory using classical Supergravity, which is a classical and low-energy limit of String Theory ($g_{YM}^2 \ll 1$ and $g_{YM}^2 N \gg 1$ limit of the aforementioned case of AdS/CFT). This route is quite effective and allows to uncover interesting dynamics of gauge theories which cannot be done using standard perturbative techniques (see this classical review of Aharony et al. on the subject). Finally, it is very interesting example of duality between theories living in different dimensions, which is believed to be inevitable consequence of the fact that String Theory describes quantum gravity among many other things. The possibility of such a duality was put forward before AdS/CFT correspondence, being based on certain quantum black holes considerations -- it is known that the entropy, which counts degrees of freedom of BH is proportional to the area of its horizon, not the volume of the interior.
