Solution(s) to dot product of vectors

For any two specific vectors, no $C$ does NOT have to be the identity matrix. For example if one of the vectors is $0$ then $C$ can be anything.

The only way to talk about $b^{-1}$ is if $b$ is one dimensional and not $0$.

First question, let $\{v_i\}$ be a basis, then $\vec{a}=\sum_i a_i v_i$ and $\vec{b}=\sum_i b_i v_i$.

Let $C_{ij}=v_i\cdot v_j$, this is for example $\delta_{ij}$ if the basis is orthonormal, but we're not assuming that.

Then by bilinearity of the dot product:

$\vec{a}\cdot \vec{b}=\sum_i\sum_j a_ib_j v_i\cdot v_j=\sum_i\sum_j a_ib_j C_{ij}$.

The $C_{ij}$ is just a collection of numbers. $C_{ij}$ has some restrictions, as all entries can't be zero, $C_{ii}>0$ if the dot product is an inner product, and so on.

For the second question consider $\vec{b}=(1,0)^T$, $D$ the zero matrix and $C=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}$,

then:

$$C\vec{b}=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}\begin{pmatrix}1\\ 0\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}=\begin{pmatrix}0 & 0\\ 0 & 0\end{pmatrix}\begin{pmatrix}1\\ 0\end{pmatrix}=D\vec{b}$$