Symmetric matrix and Hermitian matrix, unitarily diagonalizable

Yes: if $A$ is a real matrix, then it is Hermitian if and only if it is symmetric.

Similarly, if $U$ is a real matrix, then it is unitary if and only if it is orthogonal. With that said, it is indeed true that a real symmetric matrix is unitarily diagonalizable. The spectral theorem for symmetric matrices that $A$ is orthogonally diagonalizable, i.e. there exists a real diagonal matrix $D$ and an orthogonal matrix $U$ for which $A = UDU^{-1}$. Since $U$ is real and orthogonal, it follows that $U$ is unitary, which means that $A$ is also unitarily diagonalizable.

Yesi you are right. For second question, I don't know what do you mean by unitarily diagonalizable. Real symmetric matrices are diagonalizable.