For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?
You have explicit generating (acyclic) cofibrations for pretty much any model of spectra you can think of. As you point out, most begin as levelwise model structures, and so you have explicit generating (acyclic) cofibrations before left Bousfield localization, hence still have the same generating cofibrations after. You are right that the general machinery of Hirschhorn's book does not tend to give you explicit generating acyclic cofibrations, but for all the models of spectra I can think of, there is a trick that gives you the generating acyclic cofibrations. I learned this trick from Theorem 3.11 in the paper Stable left and right Bousfield localisations by Barnes and Roitzheim. The proof has a lovely trick that uses the fact that the levelwise model structure is proper. The set of generating acyclic cofibrations is $J \cup \Lambda S$, where $J$ is the old set of generating acyclic cofibrations, and $\Lambda S$ is the set of horns on $S$, where $S$ is the set of maps you're inverting. This result is the best Hirschhorn could have hoped for, but as examples in his book show, you don't get this result without the assumption of a stable, proper model category as input. Before this result of Barnes and Roitzheim, the usual idea was to build the left Bousfield localization "by hand", without reference to Hirschhorn's book. That is done in several of Mark Hovey's papers, and it works because you know you want the new fibrant objects to be the $\Omega$-spectra. When you do it that way, you often have reasonably good control over the generating acyclic cofibrations, but the approach of Barnes and Roitzheim is even better.
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
\http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf
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
http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf
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
http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf
for a discussion of this in the more general context of parametrized stable homotopy theory.