Claims About a Doubly Symmetric Function

Let $f(x)$ be an odd function on $\mathbb R$ with $f(x)=x^3-3x$ for $0\leqslant x\leqslant 2$, and suppose the graph of $f$ is centrally symmetric about the point $(2,2)$. Determine, with proof, which of the following are true.

1. $f(3)=6$. 2. $y=f(x)-5$ has three zeros. 3. $g(x)=f(x)-x$ is periodic with period $4$. 4. The segment $y=x+\dfrac{16\sqrt 3}9$ ($x\in[-2,10]$) meets the graph of $y=f(x)$ ($x\in[-2,10]$) in $6$ points.