When I add two random variables, are they added pointwise on $\Omega$, as in $X+Y(\omega):=X(\omega)+Y(\omega)$ or are they added $X+Y(\omega_1,\omega_2)=X(\omega_1)+Y(\omega_2)$ on $\Omega \times \Omega$? Going one step further, would it be correct to define the sample mean as as $\overline{X}(\omega_1,\dots,\omega_n)=\frac{X_1(\omega_1)+\dots +X_n(\omega_n)}{n}$ over $\Omega \times \dots \times \Omega$ for $X_1\dots X_n$ iid? Many stats definitions write this as
$$\overline{X} = \frac{X_1+\dots+X_n}{n}$$ which makes it unclear as to what the domainn is and I thought for the longest time the domain was simply $\Omega$.
Typically, the underlying probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is abstracted away. The answer to your question is that it depends on what the sample space $\Omega$ is. Real-valued random variables living on the same probability space, say $X$ or $Y$, are simply maps from $\Omega$ to $\mathbb{R}$. The outcomes $\omega$ in the sample space $\Omega$ are often not explicitly referred to, as long as $\Omega$ is rich enough to accommodate the desired amount of randomness.
If you had two independent random variables $X$ and $Y$, a canonical construction of the sample space is a product space: i.e. the sample points consist of tuples $\omega = (\omega_1, \omega_2)$, where each coordinate supports a particular random variable. If you had $n$ independent random variables, you could do the same thing: $\omega = (\omega_1, \omega_2, \dots, \omega_n)$. Now more generally, you could allow for a sequence of random variables with sample points $\omega = (\omega_1, \omega_2, \dots)$.
Typically, this last setting is the probability space working under the setting where you have a sequence of random variables, such as the sample mean you are asking about. In this case, we still only have the one probability space $(\Omega, \mathcal{F}, \mathbb{P})$, so $\overline{X}_n$ is still a map from $\Omega \to \mathbb{R}$.