you should have a look at Maulik's paper "Supersingular K3 surfaces for large primes" (arxiv.org/abs/1203.2889), and in particular to Section 4

actually, the canonical bundle is usually only big and nef, not ample (it is ample on the canonical model). One obvious answer: since $b_1=0$ (first Betti number), we know that the Albanese variety is trivial, i.e., every morphism to an Abelian variety/a torus is trivial. There are plenty of minimal surfaces of general type, and as far as I know, we do not have a good picture at all, even over the simply connected ones. However, being a simply connected 4-manifold, its homeomorphism type is determined by its intersection form (Freedman's theorem), but this has nothing to do with Hodge theory.

you might want to have a look at Morrison's 1987 paper "Isogenies between Algebraic Surfaces with Geometric Genus One".

Did you already have a look at Faltings' paper "Stable G-bundles and projective connections", J. Alg. Geom. 2 (1993), 507-568? I remember something about S-equivalence classes being contracted on the strictly semistable locus...

even worse: a general (algebraic) K3 surface contains infinitely many rational curves, giving examples very away from being Kobayashi hyperbolic

It's also true in positive characteristic: see the proof of Proposition (1.5) of Artin's article "Supersingular K3 Surfaces".