1- Scalar fields generally involve a coset manifold structure $G/H$, the coordinates to which basically describe scalar fields as per some sigma model structure. The manifold describing scalar fields is acted upon by $G$ in the usual sense of $G$ being the isometry group of the manifold. The vector fields involved in $\mathcal{N}=8$ with $d=4$ follow a kind of duality relation (necessary for coupling of vectors to scalar and spinors in a sigma model way) and the $28$ vector fields with their duals are assigned to $56$ pseudo-real representation of $E_{7(7)} $. So$ E_7$ which acts on scalar fields also act on vector fields through duality transformation a property shared by all extended supergravity models.
2- As said before, the coset $E_{7(7)}/SU(8)$ describes a complex manifold the coordinates to which are scalar fields.Usually, these manifolds have <PERSON> structure however in this case the manifold is an example of non-kahler manifold which arises as a result of CPT self-conjugacy of the $\mathcal{N}=8$ graviton multiplet. Scalar fields take value on this manifold and not in the vacuum of the model.
To describe the vacuum structure however, one typically gauge a subgroup of the full isometry group which should also be a subgroup of the group preserving symmetry of the bosonic part of Lagrangian. Structure of the vacuum with the popular choice of the gauge group $SO(8)$ carries a full symmetry of $Osp(4/8)$ and so the background space-time has an AdS structure. 578
$\mathcal{N}=8$ can be truncated down to lower number of supersymmetry (for model building) on the basis of the study of the critical points of the scalar potential which in turn can be determined by knowing the coset structure.
Note- Duality relations are simply symmetries of the field equations but not of Lagrangian. Here they provide an extension of symmetries of non linear sigma model used to describe scalar fields to vector fields.
These duality relations are basically a generalized form of duality symmetry of classical <PERSON> equations where exchanging E and B is a symmetry of classical <PERSON> equations in vacuum but is not a symmetry of Maxwell Lagrangian.
In extended supergravity model $\mathcal{N}\geqslant3$, the arbitrariness of sigma model metric disappears due to a change in the automorphism group of supersymmetry beyond $\mathcal{N}=3$ and for these models if we employ duality invariance then the metric structure along with G is determined completely (along with some other restrictions which determines H). Hence the coset structure is completely determined for extended SUGRA models which can be used to completely fix the interactions and hence the Lagrangian. This task was first used for $\mathcal{N}=3$ in the following reference-
Castellani, <PERSON>, <PERSON>, <PERSON>: The complete $\mathcal{N}=3$ matter coupled supergravity, Nucl. Phys. B286 (1986). 469