An $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$

Question

$A$ is an $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$. Then, which of the following is true?

1. $A^n=A^{n-2}$

2. rank of $A$ is $2$

3. rank of $A$ is atleast $2$

4. there are non-zero vectors $x$ and $y$ such that $A(x+y)=x-y$

Using Cayley-Hamilton Theorem, (1) is true. But, I can't figure out whether the other options are true or not.

I guess, (3) is also true because of the diagonal matrix $$A=D\left(\underbrace{0,0,\dots ,0}_{n-2 \text{ times}}, -1,1\right)$$ for even $n$ as in this case $$A-xI=D\left(\underbrace{-x,-x,\dots ,-x}_{n-2 \text{ times}}, -x-1,1-x\right)$$ but of course, that is not a proof.