If matter is homogeneously distributed (aka perfect fluid), there can be no spatial curvature on large scales (more precisely, homogeneity and isotropy leads to 3 cases with $k=\pm 1, 0$ with maximal spacetime symmetry). Curvature appears when you consider smaller structures, like stars and galaxies: in these cases matter is concentrated in a region of space.

Yes, k is determined from observation of density parameters, and it's very close to 0.

If matter is homogeneously distributed (aka perfect fluid), there can be no spatial curvature on large scales (more precisely, homogeneity and isotropy leads to 3 cases with $k=\pm 1, 0$ with maximal symmetry). Curvature appears when you consider smaller structures, like stars and galaxies: in these cases matter is concentrated in a region of space.

Yes, k is determined from observation of density parameters, and it's very close to 0.