for the second one by taking inverse on both sides we arive at
P^-1A^-1P = D^-1 ?
And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?
If a diagonal matrix has an inverse then the inverse is also diagonal. So all you need to do is to explain why D is invertible. (Reason: the diagonal elements of D are the eigenvalues of A and, since A is invertible, none of them can be zero.)