Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is $\overline{F(X^*)}^{wk*}$, the weak*-closure of the finite rank operators on $X^*$? Since this is rather vague, here are some concrete questions:

Q1: Do we always have $K(X^*)\subseteq\overline{F(X^*)}^{wk*}$, i.e., is every compact operator in the weak*-closure of finite-rank operators?

Q2: Is there a characterization when $B(X^*)=\overline{F(X^*)}^{wk*}$, i.e., when the finite-rank operators are weak*-dense?

Q3: Is there a characterization when $B(X^*)=\overline{K(X^*)}^{wk*}$, i.e., when the compact operators are weak*-dense?

Considering $B(X)$ as a subalgebra of $B(X^*)$ in the usual way, we may ask related questions in connection with $F(X)$ and $K(X)$. The principle of local reflexivity implies $\overline{F(X)}^{wk*}=\overline{F(X^*)}^{wk*}$ in $B(X^*)$. However, it is not clear to me if we always have $\overline{K(X)}^{wk*}=\overline{K(X^*)}^{wk*}$ (I guess not). Therefore, we may also ask:

Q4: Do we always have $K(X)\subseteq\overline{F(X^*)}^{wk*}$?