Glauber's coherent state is denoted by $\ketα$ and its two-mode extension is represented by $\ket{α,β}$. In this work, we introduce a two-mode superposition operator $A=tab+ra^\dagger b^\dagger$, whose action on the two-mode coherent state produces the two-mode coherent superposed quantum state $\ketψ=(tab+ra^\dagger b^\dagger)\ket{α,β}$. We investigate the nonclassicality and quantum non-Gaussianity of this state by means of the Wigner distribution and Wigner logarithmic negativity. Once its intrinsic nonclassical and non-Gaussian structure is established, the state is employed as the entangled resource in the Braunstein-Kimble continuous-variable (CV) teleportation protocol. We compute the ideal teleportation fidelity for coherent and squeezed inputs and analyze how the strengths of nonclassicality and non-Gaussianity influence the teleportation efficiency. Our results identify specific parameter regimes where enhanced non-Gaussian features or increased nonclassicality enable fidelities beyond the classical threshold, thereby revealing the operational significance of engineered two-mode quantum states in CV quantum information processing.