adjacency matrix of a graph and lines on quartic surfaces

Yes, this approach has been tried, and we're about to submit a paper [edit: the paper has now been submitted, see arXiv:1601.04238]. Alas, very little is known about hyperbolic (as we call them; those with a single positive eigenvalue) graphs, and currently the proof is heavily computer aided (too many cases) and still using some algebraic geometry (each line gives rise to an elliptic pencil, and these pencils are studied arithmetically). Accidentally, just the Picard number estimate is not enough: one also has to use more subtle criteria of embeddability of a lattice to $2E_8\oplus3U$.

Segre's bound is $64$ in general (there is a gap in the proof) and $48$ in your case (no plane fully split; no gap). For the moment, see arXiv:1303.1304 for further details and modern proof.

A while ago I had a long discussion of something along these lines (although the setting was somewhat different, IIRC) with colleagues, who are, unlike me, experts on K3 things. First of all, there is a bound on the number of lines on quartics due to B.Segre. Another outcome of the discussion was that there is a connection with combinatorics of generalised quadrangles (GQs) with lines of size 4. It turned out that some examples give rise to the (famous in combinatorics) examples of GQ(3,5) and GQ(3,9), with, respectively 64 and 112 points (giving the respective configurations of lines -- it escapes me now whether the points or the lines of GQs correspond to the lines on surfaces). Details escape me now. I'll email this question to these colleagues, perhaps they can comment more.

Regarding the graph eigenvalues question, there is a lot of literature. If the minimal eigenvalue of the adjacency matrix is at least -2, then you have a full classification of such graphs. See e.g. the book by Brouwer and Haemers.