Consider a sequence of random variables $\{X_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_n:\Omega \rightarrow \mathbb{R}^k$. Let $B\subseteq \mathcal{B}(\mathbb{R}^k)$ where $\mathcal{B}(\mathbb{R}^k)$ is the Borel $\sigma$-algebra over $\mathbb{R}^k$. Let $\bar{B}$ be the closure of $B$ and $B^{o}$ be the interior of $B$. Let $(\mathbb{R}^k, \mathcal{B}(\mathbb{R}^k), P)$ be the measure space induced by each $X_n$.
Can we say that
$$ \liminf_{n \rightarrow \infty} P(X_n \in B^{o})\leq \limsup_{n \rightarrow \infty} P(X_n \in B^{o})\leq \liminf_{n \rightarrow \infty} P(X_n \in B) \leq \limsup_{n \rightarrow \infty} P(X_n \in B)\leq \liminf_{n \rightarrow \infty} P(X_n \in \bar{B}) \leq \limsup_{n \rightarrow \infty} P(X_n \in \bar{B}) $$
?
This is false. Consider e.g. $X_n=0$ if $n $ even and $X_n =1$ if $n $ is odd. Now take $B=\{0\} $. Then $\limsup_n P (X_n \in B) =1 >0=\liminf_n P (X_n \in \overline {B}) $.
Similar examples apply to other inequalities of the form $\limsup_n A \leq \liminf_n B $ for the $A,B $ that you consider.