Prove that the parity operator is Hermitian

Set $x=-\xi$ in $$ \int_{-\infty}^{\infty} f(x)g(-x)\,dx $$ to get $$ \int_{-\infty}^{\infty} f(x)g(-x)\,dx=\int_{+\infty}^{-\infty}f(-\xi)g(\xi)d(-\xi)\\ = - \int_{+\infty}^{-\infty}f(-\xi)g(\xi)d\xi= \int_{-\infty}^{+\infty}f(-\xi)g(\xi)d\xi $$ so $\langle Pf,g\rangle = \langle f,Pg\rangle$

Regarding eigenvalues, notice that the parity operator is an involution, in the present context means it is it's own inverse. Next, use that every function can be expressed as the sum of its symmetric and antisymmetric part. Think that it does the job.

Please note that we assume the operator is hermitian with respect to some integration interval. For an accessible discussion on this, check Shankar's Principles of Quantum Mechanics chapter 2 or 3 I think