We consider minimization problems for nonlinear nonlocal elastic energies motivated by peridynamic-type models from continuum mechanics. The strain energy densities involve the magnitude of projected ``directional" difference quotients of the displacement. We will describe sufficient conditions on the interaction kernels for the existence of minimizers. These conditions extend and generalize previously known sufficient conditions for several different classes of nonlocal problems. Additionally, it turns out that the energy space naturally arising for a certain class of these problems is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. This equivalence permits us to apply classical Sobolev embeddings in the process of proving that critical points of the nonlocal energies enjoy both improved differentiability and improved integrability.