Interference of orbital electron wavefunctions

Inn principle it should be solved as a many-electron system, i.e. the electrons should be described by a joint wave function $Psi_{s_1,...,s_N}(mathbf{r}_1,...,mathbf{r}_N)$. However, in practice the many-body problem is no less challenging in quantum mechanics than in classical one, so one would often use approximations, representing multi-electron states as combinations of single-electron orbitals. Since these belong to different electrons, there is no interference them, although they do need to be modified to accommodate the Pauli principle and the electron-electron Coulomb interactions.

These are different electrons that are in different orbitals, all labelled by different quantum numbers and hence distinguishable. Therefore, they do not interfere.

You will still detect a higher charge density if you hover over at a position where more than one orbital has a maximum. But this is the result of the sum of the intensities $|psi_1|^2 + |psi_2|^2$ (incoherent) rather than sum of the wavefunctions (coherent -> interference) $|psi_1 + psi_2|^2$.