No, let $\mathcal F_n$ be the powerset of $\{0,1\}$.

Let us write $X\in\{0,1\}^{\mathbb N}$ as $(X_n)_n$.

Let $B=\{X:X_n=1$ for at most finitely many $n\}$. Suppose $B=$ lim sup $A_n$. Then

$X_n=1$ for at most finitely many $n$ $\iff$ infinitely many $A_n$ occur.

Case 1: for infinitely many $n$, $A_n=\{0,1\}$. Then $X_n=1$ for all $n$ is a counterexample.

I'll let you think about the other cases.

No, let $\mathcal S_0=\{\{0,1\},\emptyset\}$ and let $\mathcal S_1$ be the powerset of $\{0,1\}$. Let $\mathcal F_n=\mathcal S_0^{n-1}\times\mathcal S_1 \times\mathcal S_0^{\infty}$.

Let us write $X\in\{0,1\}^{\mathbb N}$ as $(X_n)_n$.

Let $B=\{X:X_n=1$ for at most finitely many $n\}$. Suppose $B=$ lim sup $A_n$. Then

$X_n=1$ for at most finitely many $n$ $\iff$ infinitely many $A_n$ occur.

Case 1: for infinitely many $n$, $A_n=\{0,1\}$. Then $X_n=1$ for all $n$ is a counterexample.

I'll let you think about the other cases.

No, let $\mathcal F_n$ be the powerset of $\{0,1\}$ and let $B$ be the event that $X_n=1$ for at most finitely many $n$. Then if $B$ is lim sup $A_n$ then

$X_n=1$ for at most finitely many $n$ $\iff$ infinitely many $A_n$ occur.

Case 1: for infinitely many $n$, $A_n=\{0,1\}$.

Then $X_n=1$ for all $n$ is a counterexample.

I'll let you think about the other cases.

No, let $\mathcal F_n$ be the powerset of $\{0,1\}$.

Let us write $X\in\{0,1\}^{\mathbb N}$ as $(X_n)_n$.

Let $B=\{X:X_n=1$ for at most finitely many $n\}$. Suppose $B=$ lim sup $A_n$. Then

$X_n=1$ for at most finitely many $n$ $\iff$ infinitely many $A_n$ occur.

Case 1: for infinitely many $n$, $A_n=\{0,1\}$. Then $X_n=1$ for all $n$ is a counterexample.

I'll let you think about the other cases.