Why $||(T-\lambda I)^*v||=||(T^*-\bar{\lambda}I)v||$ where $T$ is normal operator
This is simply using the following properties of the adjoint: $$(A+B)^* = A^* + B^* \tag{1}$$ $$(\lambda A)^* = \overline{\lambda} A^* \tag{2}$$ Can you prove them?
Edit. Proof of $(2)$: $$\langle x, (\lambda A)^* y\rangle = \langle \lambda Ax,y\rangle = \lambda \langle Ax,y\rangle = \lambda \langle x, A^*y\rangle = \langle x, \overline{\lambda} A^* y\rangle$$ for all $x,y\in \mathcal H$, where $\mathcal H$ is the Hilbert space you're working with.