Why does an anti-symmetric function has zero amplitude at the center of an even length window
A real symmetric function has the following property $$f(t) = f(-t)$$

whereas a real anti-symmetric function has the following property $$f(t) = -f(-t)$$

So at $t=0$ you have $$f(0) = -f(0)$$ which implies that $$f(0) = 0$$

A complex conjugate-symmetric function has the following property $$f(t) = f^*(-t)$$

whereas a complex conjugate-antisymmetric function has the following property $$f(t) = -f^*(-t)$$

So at $t=0$ you have $$f(0) = -f^*(0)$$ which implies that $f(0)$ is real and further $$f(0) = 0$$