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Given a quaternion obtained by compositions of rotations and translations, is it safe to take the $3\times 3$ upper left matrix as the "full rotation" and the last column of the matrix as the "full translation" ?
Using block-matrices, I have : $$ \left[ \begin{matrix} R & t\\ 0 & 1 \end{matrix} \right] = \left[ \begin{matrix} 1 & t\\ 0 & 1 \end{matrix} \right] \times \left[ \begin{matrix} R & 0\\ 0 & 1 \end{matrix} \right] $$ where $R$ would be the rotation and $t$ the translation.
Am I right?
Note that what you are describing are not quaternions, they are homogeneous transformations. A quaternion would be used to perform the same function as your $R$, however, you would not store it (or use it) as a matrix. Additionally, quaternions do not encode translations, only rotations.
To answer your question, yes you can decompose a homogeneous transformation into its constituent rotation and translation parts and the decomposition that you show above is correct.
