one should also mention Serre's article {\em Sur la topologie des vari\'et\'es alg\'ebriques en caract\'eristique $p$} from 1958, where the first examples of this kind were constructed.

I don't know of any reference. If I had to guess: probably the semipositivity fails if f is not generically smooth (the Albanese morphism of a surface of general type with c2<0 might provide a counter-example). If f is generically smooth, it might actually be true, except for some very few cases.

for relative dimension 1, see Szpiro's Section 3-2 of "Seminaire sur les pinceaux de courbes de genre au moins deux", asterisque 86 (1981).

yes! For a modern account, have a look at Martin Olsson: "Semistable degenerations and period spaces for polarized K3 surfaces", Duke Math. J. 125, 121-203 (2004). There, the problem is analyzed from the point of view of logarithmic geometry and you find a discussion of the toroidal point of view (plus references).

for this algorithm, we first have to find divisors on the K3 and codimension 2 cylces on the self-product of the K3 by 'going through Hilbert schemes of a suitable projective space' in order to get a lower bound on the rank. I am not an expert, but I'd be interested in how complicated this is in practice, especially, if the rank is large.

In fact, there are tricks to decide rank $1$ by reducing modulo one single prime only: arxiv.org/abs/1006.1972 - check out also the other papers of these authors circling around Picard groups of K3s

Piotr, thanks - that's a great argument!

yes, sorry, I meant supersingular. I had the naive idea of blowing up s.th. that is not F-split might give a counter-example.

I would guess that there do exist Fano varieties that are not F-split. As a possible counter-example (just a guess), I would suggest looking at the blow-up of projective 3-space along an ordinary elliptic curve.

P.S.: if you assume $X$ to be an algebraic space, and $Y$ to be an Artin stack with finite inertia, then, I think, you can argue via the coarse algebraic space of $Y$ (which exists by Keel-Mori), thereby reducing to the case that $Y$ was an algebraic space to start with. Right?

The arguments for algebraic spaces and schemes are the same. For a precise statements, see Section 4 and in particular, Theorem 4.5, of M. Artin: "Algebraic Spaces", Yale University Press (1971).