Here is my recent result (draft) that may give a better idea for Q2.

http://page.mi.fu-berlin.de/haochen/data/GraphJoin.pdf

Notations:

• $K_n$ is the complete graph on $n$ vertices
• $P_n$ is the path on $n$ vertices
• $C_n$ is the cycle on $n$ vertices
• $G_n$ is any graph on $n$ vertices
• $\lozenge_d$ is the graph of $d$-orthoplex
• $G\star H$ is graph join

I investigated on some small graphs that can be written in form of graph joins. Notably:

• $K_3\star P_6$ is not $3$-ball packable, but $K_3\star C_6$ is.
• $K_4\star P_6$ is not $4$-ball packable, $K_d\star P_5$ is not $d$-ball packable for $d\geq 5$.
• $\lozenge_{d-1}\star P_4$ is not $d$-ball packable, but $\lozenge_{d-1}\star C_4$ is.
• $G_3\star G_6$ is not $3$-ball packable with the exception of $C_3\star C_6$.
• $G_4\star G_4$ is not $3$-ball packable with the exception of $C_4\star C_4$.
• $G_4\star G_6$ is not $4$-ball packable with the exception of $C_4\star \lozenge_3$.

Finally I obtained a characterization of $3$-ball packable stacked $4$-polytopal graph:

The graph of a stacked $4$-polytope is $3$-ball packable if and only if it does not contain six $4$-cliques sharing a $3$-clique.

That means, $K_3\star G_6$ are forbidden induced subgraphs.

I recently showed that:

The graph of a stacked $4$-polytope is $3$-ball packable if and only if it does not contain six $4$-cliques sharing a $3$-clique.

While Eppstein, Kuperberg and Ziegler 2003 proved that

No stacked 4-polytope with more than 6 vertices is edge-tangent.