This is in response to the second question: the stack of all K3 surfaces is not algebraic. If you're working with Artin's axioms, then the problem is that there exist formal deformations that are not effective, i.e., one can write down a compatible system of K3 surfaces $X_n \to \mathrm{Spec}(k[t]/(t^{n+1})$ which do not come from an algebraic K3 surface over the power series ring, at least when the field $k$ is big enough. I learnt this fact from Jason Starr's paper www.math.sunysb.edu/~jstarr/papers/moduli4.pdf (see page 14 and 15).

This is in response to the second question: the stack of all K3 surfaces is not algebraic. If you're working with Artin's axioms, then the problem is that there exist formal deformations that are not effective, i.e., one can write down a compatible system of K3 surfaces $X_n \to \mathrm{Spec}(k[t]/(t^{n+1}))$ which do not come from an algebraic K3 surface over the power series ring, at least when the field $k$ is big enough. I learnt this fact from Jason Starr's paper www.math.sunysb.edu/~jstarr/papers/moduli4.pdf (see page 14 and 15).

Added later: As Scott mentions in the comments, this problem is specific to the non-polarised case. Any formal deformation of a pair (X,L) (where X is a projective variety, and L is line bundle giving a projective embedding) is automatically algebraic by formal GAGA.