Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces

Let $X$ be a~Banach space and $\mathcal{S}\mathit{eq}(X^{\ast \ast })$ (resp., $X_{{\mathrm{\aleph }}_0}$) the subset of elements $\psi \in X^{\ast \ast }$ such that there exists a~sequence $(x_n{)}_{n\ge 1}\subset X$ such that $x_n\to \psi$ in the $w^{\ast }$-topology of $X^{\ast \ast }$ (resp., there exists a~separable subspace $Y\subset X$ such that $\psi \in {\bar{Y}}^{w^{\ast }}$). Then: (i) if $\mathrm{Dens}(X)\ge {\mathrm{\aleph }}_1$, the property $X^{\ast \ast }=X_{{\mathrm{\aleph }}_0}$ (resp., $X^{\ast \ast }=\mathcal{S}\mathit{eq}(X^{\ast \ast })$) is ${\mathrm{\aleph }}_1$-determined, i.e., $X$~has this property iff $Y$ has, for every subspace $Y\subset X$ with $\mathrm{Dens}(Y)={\mathrm{\aleph }}_1$; (ii) if $X^{\ast \ast }=X_{{\mathrm{\aleph }}_0}$, $(B(X^{\ast \ast }),w^{\ast })$ has countable tightness; (iii) under the Martin's axiom $\mathit{MA}({\omega }_1)$ we have $X^{\ast \ast }=\mathcal{S}\mathit{eq}(X^{\ast \ast })$ iff $(B(X^{\ast }),w^{\ast })$ has countable tightness and $overline\text{co}(B)={\overline{\text{co}}}^{w^{\ast }}(K)$ for every subspace $Y\subset X$, every $w^{\ast }$-compact subset $K$ of $Y^{\ast }$, and every boundary $B\subset K$.