This study focused on tridiagonal Toeplitz matrices based on second Dirichlet-Neumann boundary conditions and conducted in-depth research on their eigenvalue sensitivity. After evaluating condition numbers from both individual and global perspectives using parameters, our framework further achieved spectral sensitivity assessment in the context of second Dirichlet-Neumann tridiagonal Toeplitz matrices. The analysis process examined the $\epsilon$-pseudospectra. Finally, we explored an inverse eigenvalue problem embedded in a constrained optimization framework, where the trapezoidal second Dirichlet-Neumann tridiagonal Toeplitz matrix generated by this problem becomes the ultimate optimal computational solution.