From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ together with the $\sigma-$algebra $\mathcal{F}$ generated by cylinder functions forms a measurable space $(\Omega, \mathcal{F})$. We define a probability measure $P$ on $(\Omega, \mathcal{F})$ and a time homogeneous Markov chain $\{X_n\}_{n \ge 0}$ with transition probabilities \begin{equation} Q_{\omega} (X_{n+1} = k|X_n=v) =\omega_v (k) \end{equation} $Q_{\omega}$ is called the quenched law of the RW, and the marginal of $\mathbb{P}:=P\otimes Q_{\omega}$ on $V^{\mathbb{N}}$ is the annealed law of the RW. I intentionally used the letters $Q$ and $k$ above.

1. Is this what I should understand from this definition:

$\omega$ is simply an element of the set $\Omega$, and $P$ is a (probability) measure on $(\Omega, \mathcal{F})$. Using $\Omega$ as an index set, $Q_{\omega}$ is the probability measure on $(E_{\omega}, \Sigma_{\omega})$ and $X_n=X_n(\omega, z)$ is a $V-$valued measurable function, i.e., $X_n: (E_{\omega}, \Sigma_{\omega}, Q_{\omega}) \to (V,\mathcal{S})$ for each $\omega \in \Omega$.

2. The environment $\omega$ can be viewed as a sequence of random variables. How can I connect these two definitions?

Thanks for your time and comments in advance.