Is a negated positive (semi-)definite matrix always negative (semi-)definite and vice versa? If Matrix $A$ is positive definite. Does it hold for every $A$ that $-A$ is negative definite? Does the same hold for positive semidefinite $A$ too, that $-A$ is negative semidefinite? Thank you.

Yes. Because (for instance) $x'(-A)x = -x'Ax$, so any vector $x$ that is a witness for or against the positive definiteness of $A$ is also a witness for or against the negative definiteness of $-A$, and so on.