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Here is a thought on the first question. What you need to know (at least to get an algebraic space; I'll let others be more careful than I if you want a scheme) is how large n must be to ensure that an automorphism of a smooth genus g curve X which fixes n points must be the identity. Let G be the cyclic group generated by this automorphism: then the map X -> X/G is totally ramified at your n fixed points. So by Riemann-Hurwitz, g(X) is at least -2|G| + n(|G|-1). If G is nontrivial, in other words, g is at least n-4. So I think g+5 marked points should be enough. That this is necessary can be seen by taking g=2; on M_{2,6} you'll have a bunch of loci with an extra involution, parametrizing curves whose marked points are precisely the Weierstrass points.