Motivated by the state of the art method for fabricating high-density periodic nanoscale defects in graphene, the structural, mechanical, and electronic properties of defect-patterned graphene nanomeshes including diverse morphologies of adatoms and holes are investigated by means of first-principles calculations within density functional theory. It is found that various patterns of adatom groups yield metallic or semimetallic, even semiconducting, behavior and specific patterns can be in a magnetic state. Even though the patterns of single adatoms dramatically alter the electronic structure of graphene, adatom groups of specific symmetry can maintain the Dirac fermion behavior. Nanoholes forming nanomesh are also investigated. Depending on the interplay between the repeat periodicity and the geometry of the hole, the nanomesh can be in different states ranging from metallic to semiconducting including semimetallic states with the bands crossing linearly at the Fermi level. We showed that forming periodically repeating superstructures in a graphene matrix can develop a promising technique for engineering nanomaterials with desired electronic and magnetic properties.