Let's define a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and consider a nonnegative random variable $Y$ defined on it such that $Y\in\mathcal{L}^{1}$.
Since, for a constant $c$, it always holds true that: \begin{equation} Y \mathbb{1}_{\{Y>c\}}\leq Y \end{equation} am I allowed to apply Lebesgue's Dominated Convergence Theorem to state that: \begin{equation} \lim\limits_{c\rightarrow \infty} \mathbb{E}\{Y \mathbb{1}_{\{Y>c\}}\}=\mathbb{E}\{\lim\limits_{c\rightarrow\infty} Y \mathbb{1}_{\{Y>c\}}\} = 0 \end{equation} $\big($where the last equation follows from the fact that $\lim\limits_{c\rightarrow\infty} Y \mathbb{1}_{\{Y>c\}}=0$ a.s., hence $\mathbb{E}\{\lim\limits_{c\rightarrow\infty} Y \mathbb{1}_{\{Y>c\}}\}=\mathbb{E}\{0\}=0\big)$?
I think I am allowed to do so, but I would like to have a confirmation about that
Yes, it works, but dominated convergence only works for sequences.
So actually, you should show for every sequence $(c_n)_n$ with $c_n \to + \infty$ that
$$\lim_n \mathbb{E}[Y1_{\{Y > c_n\}}] = 0$$
in order to conclude that
$$\lim_{c \to \infty} \mathbb{E}[Y1_{\{Y > c\}}]= 0$$