Yes, this is provable in NBG. To see this, let $F$ be a one-one function from $\mathcal P(A)$ into $A$. By induction we define a one-one function $G$ from $V$ into $A$ as follows. Assume $G$ is defined on $V_\alpha$, and suppose that $x\in V_{\alpha+1}\setminus V_\alpha$. Then we let $G(x) = F(G[x])$.

Yes, this is provable in NBG. To see this, let $F$ be a one-one function from $\mathcal P(A)$ into $A$. By transfinite recursion on $\in$, we define a function $G$ from $V$ to $A$ such that $G(x) = F(G[x])$. A simple induction then establishes that $G$ is one-one.