Consider the following model $(X,F, P_a:a \in A)$ where $X$ is the sample space, F the corresponding sigma-algebra and $P_a$ a probability measure for given parameter $a \in A$. Assume $P_a << \mu$. Therefore there exists a denisty $q_a$ w.r.t. $\mu$. We consider the product model of the given model: $(X^n, F^{ \otimes n}, P^{\otimes n}_a:a \in A)$. Therefore the density can be written as $\prod_{k=1}^n q_a(x_k)$
Consider $(\ln q^{\otimes n})(X) = \sum_{k=1}^n \ln q(X_k) $
How can I see that $( \ln q(X_k))_{k=1}^n$ is i.i.d. w.r.t. $P^{\otimes n}_a$ Is this trivial because the product space is constructed in that way s.t that the density factorizes? Thus for a random variable $X$ holds: $X: X^n\rightarrow X^n$, s.t. $X_k(x_1,..,x_n) := x_k$
Why can I let $q^{\otimes n}$ depend on a random variable $X$. How does this make sense?
You should be a bit careful with using $X$ to denote both the space and a random vector. Therefore, let $(\mathcal{X},\mathcal{F})$ denote the measurable space. For each $k\in\{1,\dotsc,n\}$ we let $X_k\colon \mathcal{X}^n\to \mathcal{X}$ be given by $X_k(x_1,\dotsc,x_n)=x_k$. A consequence of equipping the measurable space $(\mathcal{X}^n,\mathcal{F}^{\otimes n})$ with the probability measure $P_a^{\otimes n}$ is that the random variables $X_1,\dotsc,X_n$ will be i.i.d. with $X_1\sim P_a$. Indeed, for any $A_1,\dotsc,A_n\in\mathcal{F}$ $$ P_a^{\otimes n}(\bigcap_{k=1}^n\{X_k\in A_k\})=P_a^{\otimes n}(\{x\in\mathcal{X}^n|x_1\in A_1,\dotsc,x_n\in A_n\})=P_a^{\otimes n}(A_1\times\dotsm\times A_n)=\prod_{k=1}^nP_a(A_k). $$ Note that $q_a$ and $q_a^{\otimes n}$ are non-negative functions defined on $\mathcal{X}$ and $\mathcal{X}^n$ respectively. Hence, it is perfectly valid to consider e.g. $q_a^{\otimes n}(X)=\prod_{k=1}^n q_a(X_k)$ which is then a non-negative function defined on $\mathcal{X}^n$, i.e. a non-negative random variable.
Finally, since the random variables $X_1,\dotsc,X_n$ are i.i.d. under $P_a^{\otimes n}$ it follows that $\ln q_a(X_1),\dotsc,\ln q_a(X_n)$ are i.i.d. under $P_a^{\otimes n}$.