This is only an answer to the second question: Yes, a square complex matrix is always $*$-congruent to its transpose, according to a more general result proved by Horn and Sergeichuk in "Congruences of a square matrix and its transpose". They prove the result for all fields with involution in characteristic other than 2.

An answer to the second question: Yes, a square complex matrix is always $*$-congruent to its transpose, according to a more general result proved by Horn and Sergeichuk in "Congruences of a square matrix and its transpose". They prove the result for all fields with involution in characteristic other than 2.

Added: Here's a counterexample for your first question:

$\left( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right) \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right) \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right) = \left( \begin{matrix} 0 & 1 \\ 0 & 1 \end{matrix} \right)$.