The first question is:Assume we have four random variables $A,B,C,D$, each one is a positive random variable with support $\text{supp}X$ but they are not independent. Given that if $A,B$ are given, the random variable $C$ is drawn from its marginal distribution $P_C$. Furthermore, given that $D$ and $C$ are not independent and $D$ is also not independent with both $A$ and $B$. I don't know whether $C$ and $D$ are conditionally independent given $A,B$ or not.
Hence, the question is: whether the following statement is true or not$$E[C\vert A,B,D]=E_{P_C}[C|A,B]?$$
The second related question is: If we assume $A$ is $F$-measurable and $P_A(A=x|F)$ is a $pmf$ for all support $x\in\text{supp}P_A$ and $B$ is $G$-measurable where $F$ and $G$ are two sigma-algebra and $F\subset G$. Given that $A$ and $B$ are not independent. The question is: whether the following statement is true or not$$P_A(A=x|F,B)=P_A(A=x|F)$$ where $x$ is in the support of the distribution $P_A$.