Altermagnetic $\alpha$-MnTe with N'eel vector along the $y$-axis exhibits a finite anomalous Hall conductivity (AHC) and weak ferromagnetism along the $z$-axis. As already demonstrated in the bulk, there is a breaking of the C${}_6$ symmetry by the in-plane N'eel vector, leaving a C${}_2$-type magnetic symmetry. The (001) surface of $\alpha$-MnTe breaks the C${}_2$, leaving only a time-reversed mirror symmetry with respect to the $x=0$ plane. Therefore, we demonstrate that, on the surface, the interplay between the breaking of crystal symmetry and N'eel vector orientation reduces the space group from hexagonal P6${}_3$/mmc to orthorhombic Amm2. As a result, the surface exhibits not only a polar distortion along the $z$-axis but also along the $y$-axis. Motivated by this, we separately examine the effect of the in-plane electric field in bulk MnTe. Moreover, as a doped ionic semiconductor, the properties of MnTe can be influenced by lattice polarization under an applied electric field. We investigate the interplay between the intrinsic anomalous Hall effect and lattice polarization, showing that polarization effects can substantially affect the AHC. Since the electric field breaks inversion symmetry, this contribution from the lattice polarization coexists with the nonlinear anomalous Hall effect, highlighting the rich transport phenomenology of altermagnets.