here's a try:
(edit, deleted my development for q1 which was wrong, my bad!, thanks to Henning Makholm)
- If I'm not mistaken, you can show that the determinant is positive (because I think that you can show something like $\det(A\overline{A})=\det(A)det(\overline A)=\overline{\det A}^2$) which thus implies that the product of eigenvalue be positive. The product of $\lambda$ and $\overline{\lambda}$ is obviously positive if $\rm{Im}\lambda\neq 0$ but if you have a $\lambda\in\mathbb R$ and \lambda<0 then there must be another $\lambda'\in\mathbb R$ with \lambda'<0. This implies that there be an even number of real negative eigenvalues.
(edit, as Beni mentioned, this just shows that the total number of real negative eigenvalues is even (product is positive) which might not answer the question)