Is it true that any positive matrix $\hat{H}$ can be rewritten in terms of pauli operators ($\sigma_0, \sigma_1, \sigma_2, \sigma_3$) as: \begin{align} \hat{H} = \sum_{i,j}c_{ij} \sigma_i \otimes \sigma_j \end{align} If yes, where can I find any derivation for this? Does the above relation have a specific name I can look for?
Moreover, assuming the above relation to hold true, is it correct to say that: $$c_{ij} = tr(\sigma_i \otimes \sigma_j \hat{H})$$
This is just a consequence of the fact that the set $\{\sigma_i\otimes \sigma_j\}_{i,j=0,\ldots,3}$ is a basis of the real vector space of hermitian $4\times4$ matrices. This vector space is equipped with an inner product such that for two elements $\alpha$ and $\beta$ we have that $$(\alpha,\beta) \equiv \frac{1}{4} \mathrm{Tr}(\alpha\,\beta) \quad. $$ Any hermitian $4\times4$ matrix $H$ can thus be expanded as a real linear combination of the basis states:
$$H= \sum\limits_{ij}c_{ij}\,\sigma_i\otimes \sigma_j \quad, $$
where $$c_{ij}= (\sigma_i\otimes\sigma_j,H) \quad .$$
As an exercise you could verify this by explicit calculations. The relations
$$\left(A \otimes B\right) \left(C\otimes D\right) =AC \otimes BD $$
and $$\mathrm{Tr}(A \otimes B) = \mathrm{Tr}(A)\, \mathrm{Tr}(B) $$
could be useful.
Finally, since a positive (semi-definite) matrix is by definition hermitian, we can express every positive $4\times 4$ matrix in the same fashion as above.
You might be also interested in in this Math SE post or this QC SE post, where more general cases are discussed.