Identities for $\operatorname E\left[f(X_1,X_2)\mid\mathcal F\right]$ when $X_1$ is $\mathcal F$-measurable or independent of $\mathcal F$

Question

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
• $(E_1,\mathcal E_1)$ be a measurable space;
• $E_i$ be a $\mathbb R$-Banach space and $X_i$ be an $E_i$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$ for $i=2,3$;
• $\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $\Omega$;
• $f:E_1\times E_2\to E_3$ be $(E_1\otimes\mathcal B(E_2),\mathcal B_3)$-measurable.

Assuming $\operatorname E\left[\left\|f(X_1,X_2)\right\|_{E_3}\right]<\infty$ are we able to derive identities for $\operatorname E\left[f(X_1,X_2)\mid\mathcal F\right]$ when

1. $X_1$ is $\mathcal F$-measurable;
2. $X_1$ is independent of $\mathcal F$?

I've got an intuitive idea how these identities should be, but I don't know how to prove them. For 1., we should have $$\operatorname E\left[f(X_1,X_2)\mid\mathcal F\right]=f\left(X_1,\operatorname E\left[X_2\mid\mathcal F\right]\right)\;\;\;\text{almost surely}\tag1$$ and for 2., $$\operatorname E\left[f(X_1,X_2)\mid\mathcal F\right]=\left.\operatorname E\left[f(X_1,x_2)\right]\right|_{x_2\:=\:\operatorname E\left[X_2\mid\mathcal F\right]}\;\;\;\text{almost surely}\tag2.$$ Are we able to prove them?