Prove that the map $\phi:S^3\times S^3\to{\bf GL}(4,\Bbb R)$ defined via quaternions as$\phi(p,q)(v)=pvq^{-1}$ has image ${\bf SO}(4)$

One important thing that comes out of this argument is that $SO(4)$ almost factors as a direct product, namely its universal (double) cover Spin$(4)$ factors as the direct product of Spin$(3)$ (which is $S^3$, or equivalently $SU(2)$) with itself.

So regarding question (3), in some sense your are asking whether there is a way to see that $SO(4)$ should factor in this way. One way to see it is to consider the Dynkin diagram for the Lie algebra $\mathfrak{so}_4$; it is the Dynkin diagram $D_2$, which is the disjoint union of two points; this shows that $\mathfrak{so}_4$ factors as the direct product of two copies of $\mathfrak{so_3}$, which is the Lie algebra analogue of the direct product factorization of the spin groups.

For larger $n$ there is no such factoriation: the Lie algebra $\mathfrak{so}_n$ is simple if $n > 4$, and the associated Dynkin diagram is connected.

What comes next is not really an answer to your question, but may be of interest: the fact that $SO(4)$ (almost) factors, while other $SO(n)$ don't, is one of the reasons that $4$-dimensional geometry has such a different flavour (e.g. Donaldson's theory) compared to geometry in other dimensions.

Finally, this discussion treats some of the material in your post from a different perspective, and so may be of interest in relation to your question (2).