Suppose that you initially use upper case letters to denote r.v.s. For example, you could say
Consider the r.v.s $X_1, \dots, X_n$ and $Z_1, \dots, Z_m$ where the $Z_i$s give rise to $X_j$s
Would it make sense to use then $$\mathbb{E}\left[ p(x_1, \dots, x_n \mid z_1, \dots, z_m ) \right]$$? In other words, would these notations be consistent with each other? And why?
Or should I say instead
Consider the r.v.s $x_1, \dots, x_n$ and $z_1, \dots, z_m$ where the $z_i$s give rise to $x_j$s
that is, with lower case letters? And why?
Or maybe I should use the notation
$$\mathbb{E}\left[ p(X_1, \dots, X_n \mid Z_1, \dots, Z_m ) \right]$$
if I say
Consider the r.v.s $X_1, \dots, X_n$ and $Z_1, \dots, Z_m$ where the $Z_i$s give rise to $X_j$s
???
NOTE: I assume that the expected value operator only gets r.v.s as input. ALSO, my goal is to have consistent notation, i.e. if I use upper case letters to denote r.v.s in one place, I want ALWAYS to use upper case letters to denote r.v.s
Mathematical writing is case (and font) sensitive. This allows one, for instance, to have a collection of sets, $\mathcal{A}$, an element $A$ of $\mathcal{A}$ and an element $a$ of $A$. The three objects $a, A, \mathcal{A}$ are different. We use the same letter to remind ourselves that there is a relationship between them. So we might say $a \in A, b \in B$ but saying $a \in B, b \in A$ is weird.
In probability, one often (but not always) uses upper case letters to denote random variables and lower case letters to denote real numbers (or other non-random values). If you are using this convention, then $p(x_1,\dots,x_n|z_1,\dots,z_m)$ is a constant and hence $$\mathbb{E}[p(x_1,\dots,x_n|z_1,\dots,z_m)] = p(x_1,\dots,x_n|z_1,\dots,z_m) $$ in the same way that $\mathbb{E}[7] = 7$.
If you had defined your random variables with upper-case letters, I would most likely assume that $x_1,\dots,x_n,z_1,\dots,z_m$ are non-random values associated to the random variables for example maybe the relationship is $\mathbb{P}(X_i = x_i) = 1/2$. But before that, I would be confused why $x_1,\dots,x_n,z_1,\dots,z_m$ haven't been defined.
Otherwise, you should define your random variable as $X$ and write $\mathbb{E} [X]$ or define it as $x$ and write $\mathbb{E}[x]$ but you should not mix upper and lower cases for the same object.