If T is normal then it has a unitary diagonalisation. In other words, there is an orthonormal basis of V with respect to which T has a diagonal matrix (whose diagonal entries are the eigenvalues of T, but you don't need to know that for this question). Then the matrix of T* is also diagonal, and its diagonal entries are the complex conjugates of those of T. So you just need to show that there exists a polynomial taking a given finite set of complex numbers to their conjugates. If you don't know how to do that, do a Google search for "polynomial interpolation".

This follows quite easily from 1. If S commutes with T, then it's easy to check that S commutes with any polynomial in T.