Let $L>0$ fixed. Consider the space $$\mathcal{P}:=C_{per}^{\infty}([0,L])=\{f: \mathbb{R} \longrightarrow \mathbb{C}\; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.$$
Consider the Sobolev Space $H^1_{per}([0,L])$. This space can be interpreted as the set of $f \in \mathcal{P}'$ such that $$f, f' \in L^2_{per}([0,L]),$$ with norm $$||f||_{H^1_{per}}=(||f||^2_{L^2_{per}}+||f'||^2_{L^2_{per}})^{\frac{1}{2}}, \; \forall \; f \in H^1_{per}([0,L]), \tag{1}$$ where $$L^2_{per}([0,L])=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is periodic with period} \; L \; \text{and} \; f|_{[0,L]} \in L^2([0,L])\} $$ with inner product $$(f,g)_{L^2_{per}}=\int_0^L f(x) \overline{g(x)}\; dx, \; \forall \; f,g \in L^2_{per}([0,L]).$$
Question 1. I can rewrite $ (1) $ as $$||f||_{H^1_{per}}=(||f||^2_{L^2_{per}}+||\nabla f||^2_{L^2_{per}})^{\frac{1}{2}}?$$
Question 2. Similarly to Sobolev Space $H_0^1(\Omega)$, with $\Omega \subset \mathbb{R}^n$ open and bounded, using the Poincaré Inequality can I define a norm in $H^1_{per}([0,L])$ by $$||f||_{H^1}:=||\nabla f||_{L^2_{per}}, \forall \; f\in H^1_{per}([0,L])?$$
And in this case a inner product in $H^1_{per}([0,L])$ by $$(f,g)_{H_1}=(\nabla f, \nabla g)_{L^2}, \; \forall \; f,g \in H^1_{per}([0,L])?$$
The answer to the first question is yes, because $$\nabla f=\left(\frac{\partial f}{\partial x_1}, \dots,\frac{\partial f}{\partial x_n} \right)$$ where $n$ is the space dimension. In your case, $\nabla f=\frac{\partial f} {\partial x_1}$. Since in dimension $1$ you have only one direction you can write $\frac{\partial f}{\partial x_1}=\frac{d f}{dt}=f'$.
As for question $2$, for the Poincaré inequality to be valid, you need more hypotheses about the function $f$, for example, that the mean of it is $0$. So, for function $f \in H^{1}_{per}([0,L])$ where Poincaré's inequality fails, you will not be able to prove that $(\nabla f, \nabla f)=0 \Rightarrow f=0$