The following is mostly bogus, based on my overly quick reading and misunderstanding of the question:
No. $E_7 / (SU(8)/\mu_2)$ (where $E_7$ here denotes the compact real Lie group) is not a Hermitian symmetric space. You can tell, because there is no torus (copy of $U(1)$) in the center of $SU(8)$ (maybe Helgasson or Wolf is a good reference here).
The following comment does not apply, since the manifold has dimension $4k+2$ and not $4k$
Hence, by Gauduchon, Moroianu, Semmelmann (Inventiones Math., 2010), as you seem to know, this exceptional inner symmetric space carries no almost complex structure.
The following is correct:
I have no idea how to answer this question.