Is it formally correct to write $E[X\mid Y=y, Z]$ for the conditional expectation of $X$ given $Y=y$ and $Z$?
I know it's a stupid question but I am not sure if it is formally correct to have a value for one of the conditioning variable but not for the other.
If $A$ is an event, then $E[X \mid A] = E[1_A X] / P(A)$ where $1_A$ is the indicator random variable for the event $A$.
If $P(Y=y) \ne 0$, then the quantity $E[X \mid Y=y]$ is then a well-defined [non-random] quantity, by taking $A = \{Y=y\}$ in the above definition.
Then if $Y$ is a discrete random variable, we define $E[X \mid Y]$ as the random quantity $f(Y)$, where $f(y) := E[X \mid Y=y]$ is a [non-random] function.
Similarly, if $Y$ and $Z$ are discrete random variables, then $E[X \mid Y=y, Z]$ can be defined as $f(z)$ where $f(z) := E[X \mid Y=y, Z=z]$ is a [non-random] function.
The above can be generalized in the usual way to handle cases where $Y$ and $Z$ are not discrete random variables; just carefully follow how conditional expectations are defined in your textbook/notes.