Let $\Omega \subset \mathbb{R}^n$ open and $T>0$ fixed. Let $f \in L^2(0,T; L^2(\Omega))$.
Question 1. Can I to afirmate that there exists $c>0$ such that $$\int_0^T ||f(t)||^2_{L^2} \; dt<c?$$
Here, $||\cdot||_{L^2}$ denote the usual norm in $L^2(\Omega)$. I think yes, since the function $g:(0,T) \longrightarrow \mathbb{R}$ given by $$g(t):=||f(t)||_{L^2},\; \forall \; t \in (0,T)$$ is such that $g \in L^2(0,T)$, that is, $$\int_0^T ||f(t)||^2_{L^2} \; dt=\int_0^T |g(t)|^2_{L^2} \; dt <\infty$$
Question 2. I know that $L^2(0,T, L^2(\Omega))$ is continuous embedded in $L^1(0,T, L^2(\Omega))$. So, $g \in L^1(0,T)$, that is, $f \in L^1(0,T, L^2(\Omega))$? Moreover, $$\int_0^T ||f(t)||_{L^2} \; dt<c?$$
Where $c>0$ is the same given in Question $1$.
Definition. Let $X=(X,||\cdot||_X)$ be a Banach space and $1 \leq p < \infty$. The space $L^p(0,T; X)$ consists of all measurable functions $u: (0,T) \rightarrow X$ for which $$||u||_p:=\left ( \int_0^T ||u(t)||_X^p\;dt\right)^{\frac{1}{p}}< \infty $$ holds.
The Question 1 is true. Indeed, from the definition of the norm in the space $L^p(0,T;X)$ with $p=2$ and $X=L^2(\Omega)$, we obtain $$\left( \int_{0}^{T} \|u(t)\|_{L^2(\Omega)}^{2} dt \right)^{\frac{1}{2}}=\|u\|_{L^2(0,T;L^2(\Omega))}<\infty \Rightarrow \int_{0}^{T} \|u(t)\|_{L^2(\Omega)}^{2} dt<c<\infty.$$
To Question 2, observe that from Hölder's inequality (assuming that $ \displaystyle \int_{0}^{T} \|u(t)\|_{L^2(\Omega)}^{2} dt<c$) we obtain \begin{eqnarray} \int_{0}^{T} \|u(t)\|_{L^2(\Omega)} dt&=&\int_{0}^{T} 1 \cdot\|u(t)\|_{L^2(\Omega)} dt \\ &\leq& \left(\int_{0}^{T} 1^2dt\right)^\frac{1}{2} \left( \int_{0}^{T} \tag{*} \|u(t)\|_{L^2(\Omega)}^{2} dt \right)^{\frac{1}{2}}\\ &<& T^{\frac{1}{2}}c^{\frac{1}{2}}. \end{eqnarray}