David has answered 1-3, and I agree with him in the abstract. However, I would like to say more and specifically address 4, since there is a huge difference between the fibrant objects in the two main styles of explicit point-set level categories of spectra. Just as for the usual Quillen model structure on spaces, in the Lewis-May or EKMM categories, the fibrations are the Serre fibrations, so they are defined directly in terms of standard lifting properties and therefore EVERY object is fibrant. The cofibrant objects are just the retracts of cell spectra, defined almost exactly as in the category of spaces, so here the model structures are just like the usual model structure on spaces, as is the theory of CW spectra. The generating acyclic cofibrations are also just like in spaces. In EKMM, this is all still true for modules over a ring spectrum. See Section VII.5 of EKMM. The link to EKMM in the Question is incorrect. The correct link is
For diagram spectra (naive, symmetric, or orthogonal), the fibrant objects are $\Omega$ spectra, and to make that true one must expand the set of generating acyclic cofibrations, as is made precise in
Section 9.4 of
That gives an explict description of the generating acyclic cofibrations $K$ and Proposition 9.5 says exactly what conditions must be satisfied for a map to satisfy the right lifting property with respect to $K$.
Incidentally, there is no explicit published CW theory for diagram spectra, as far as I know, the point being that one must pay attention to the difference just described. See Section 24.1 of
for a discussion of this in the more general context of parametrized stable homotopy theory.