Numerical study is conducted to grasp the flame structure and NO emissions for a wide range of oxy-fuel combustion (covering from air blown combustion to pure oxygen combustion) and for various mole fractions of recirculated $CO_2$ in $CH4-O_2/N_2/CO_2$ counterflow diffusion flames. Special concern is given to the difference of the flame structure and NO emissions between air blown combustion and oxy-fuel combustion w/o recirculated $CO_2$ and is also focused on chemical effects of recirculated $CO_2$. Air blown combustion and oxy-fuel combustion w/o recirculated $CO_2$ are shown to be considerably different in the flame structure and NO emissions. Modified fuel oxidation reaction pathways in oxygen-enriched combustion are provided in detail compared to those in air blown combustion w/o recirculated $CO_2$. The formation and destruction of NO through Fenimore and thermal mechanisms are also compared for air blown combustion and oxyegn-enriched combustion w/o recirculated $CO_2$, and the role of the recirculated $CO_2$ and its chemical effects are discussed. Importantly contributing reaction steps to the formation and destruction of NO are also estimated in oxygen-enriched combustion in comparison to air blown combustion.