Regarding the definition of a periodic function

Question

In a local textbook, a real function of the real variable $x$ is said to be periodic if there exists a positive real $T$ such that

$$(\forall x \in D_f) \quad : \quad \begin{cases}x + T \in D_f \\ \\ f(x + T) = f(x) \end {cases}$$

What's confusing to me are two things the authors mention afterwards :

1. They drew the corollary that for all $x \in D_f$, for all $\mathbf{n \in \underline{\mathbb{Z}}}$ $f(x+nT)=f(x)$
2. For a $T$-periodic function, it suffices to study it on $D_f \cap [0,T]$.

My question regarding point (1) :

How do we prove $f(x+nT)=f(x)$ for $n \in \underline{\mathbb{Z}^-}$ while we haven't adopted the definition that imposes the further information $x-T\in D_f$ in the definition.

My question regarding point (2) :

Isn't it a redundance to use $[0,T]$ instead of $[0,T[$ while we know that

$0 \in D_f \iff T \in D_f $ and $f(0) = f(T)$ when $0\in D_f$.

Thanks for any clarifications.