The opening of an energy gap and generating magnetism in graphene are certainly the most significant and urgent topics in your current research. The majority of proposed applications for it require the ability to modify its electronic structure and induce magnetism in it. Here, using first-principles calculations utilizing the PBE and HSE06 functionals, we examine the structural, energetic, electronic, magnetic, and phonon transport characteristics of armchair graphene and boron nitride nanoribbons (aGNRs and aBNNRs), and zigzag graphene and boron nitride nanoribbons (zGNRs and zBNNRs) with varying widths. We shall emphasize the impact of incorporating aBNNRs and zBNNRs of varying widths into graphene monolayers (GMLs). The findings suggest that zBNNRs are easier to insert into GMLs than aBNNRs. A study of the average formation energies of graphene and boron nitride nanoribbons reveals that BNNRs have a formation energy that is at least twenty times greater than GNRs. We have observed energy gaps that can be classified into three distinct families in aGNRs, aBNNRs, and aBNNRs inserted into GML. In the zGNRs and zBNNRs inserted in GML, depending on the width, different magnetic orderings (antiferromagnetic, ferrimagnetic, and ferromagnetic), and electronic behaviors are observed (metallic, semimetallic, semiconductor, and topological insulator).