Consider a task of calculating the basis coefficients of a vector belonging to a tensor product of two Hilbert spaces.
Some definitions first. Take two Hilbert spaces $(A,(\cdot,\cdot)_A)$ and $(B,(\cdot,\cdot)_B)$ as they usually defined in quantum computing textbooks. For $A$, $|a\rangle \in A$ is a generic vector and $\{|a_i\rangle\}$ (variable-indexed) are basis vectors, so $\forall |a\rangle \exists \alpha_i : |a\rangle = \sum_i\alpha_i|a_i\rangle$. Note that $(|a_i\rangle,|a_j\rangle)_A = \langle a_i|a_j\rangle = \delta_{ij}$ ($1$ if $i=j$, $0$ otherwise). Same things for $B$.
Now consider a Hilbert space tensor product $(A\otimes B,(\cdot,\cdot)_{AB})$ where the mapping $\otimes : (A \times B)\to(A\otimes B)$ is bilinear and $$((a_1\otimes b_1),(a_2\otimes b_2))_{AB} \triangleq (a_1,a_2)_A\cdot(b_1,b_2)_B$$
We know that $a_i\otimes b_j$ (spelled as $|a_ib_j\rangle$) is a basis in $A\otimes B$ so for any $a\otimes b $ (aliased as $|ab\rangle$) there exist $\gamma_{ij}$ such that $|ab\rangle = \sum_{ij}\gamma_{ij} |a_i b_j\rangle$. How to calculate these $\gamma$? A simple approach would be to use the inner product as defined above. Here we go: the coefficient near the $|a_pb_q\rangle$ basis vector is:
$$ \gamma_{pq} = \langle a_p b_q|ab\rangle =_1 \langle a_p b_q|\sum_{ij}\alpha_i \beta_j |a_ib_j\rangle =_2 \sum_{ij}\alpha_i\beta_j\delta_{ip}\delta_{jq} = \alpha_p\beta_q $$ where $=_1$ follows from the bilinearity of $\otimes$, and $=_2$ follows from the definition of $(\cdot,\cdot)_{AB}$.
My questions are:
Is it correct to say that the above formula still does not define the Kronecker product of matrix representations of $|a\rangle$ and $|b\rangle$ because we didn't specify the order of basis coefficients in the corresponding matrices?
Is it the only possible way to derive $\gamma$? Do we have any reasons to assume that $\gamma_{pq} = \alpha_p\beta_q$ without referring to $(\cdot,\cdot)_{AB}$ i.e. from the fact that $A\otimes B$ is a valid tensor product plus maybe some other facts?
Why do we use the $(\cdot,\cdot)_{AB}$ as defined above in quantum computing? Is it a direct result of physical experiments or there are pure mathematical reasons to do so?