Regarding the first question, Frenet-Serret formulas don't tell anything about the initial position, or the initial orientation, of the curve they determine locally. This is exactly the difference between an ODE and an IVP; without initial conditions one can only consider the (existence and uniqueness) of the general solution. In order words, if one only knows $\kappa$ and $\tau$ of a curve in $\mathbb{R}^3$, one can't determine where the curve is, or if it's coming closer to a predetermined destination.
Regarding the second question, as Chappers commented one can transform the Frenet-Serret ODE with the unknown taking values in $M_3(M_{3\times 1}(\mathbb{R}))$ (where for $V$ a vector space $M_{n\times m}(V)$ denotes the vector space of $n\times m$ matrices whose entries are vectors in $V$) to an ODE with the unknown taking values in $M_{9\times 1}(\mathbb{R})$, as $M_3(M_{3\times 1}(\mathbb{R}))\cong M_{9\times 1}(\mathbb{R})$ as vector spaces. Note that here there is some choice involved; e.g. the isomorphism
$$\phi: \begin{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\\\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}\\\begin{pmatrix}z_1\\z_2\\z_3\end{pmatrix}\end{pmatrix}\mapsto \begin{pmatrix}x_1\\x_2\\x_3\\y_1\\y_2\\y_3\\z_1\\z_2\\z_3\end{pmatrix}$$
transforms the Frenet-Serret ODE into
$$\left(\begin{array}{c}T_1\\T_2\\T_3\\\hline N_1\\N_2\\N_3\\\hline B_1\\B_2\\B_3\end{array}\right)'=
\left(\begin{array}{ccc|ccc|ccc}
0 & 0 & 0 & \kappa & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \kappa & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \kappa & 0 & 0 & 0 \\\hline
-\kappa & 0 & 0 & 0 & 0 & 0 & \tau & 0 & 0 \\
0 & -\kappa & 0 & 0 & 0 & 0 & 0 & \tau & 0 \\
0 & 0 & -\kappa & 0 & 0 & 0 & 0 & 0 & \tau \\\hline
0 & 0 & 0 & -\tau & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -\tau & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -\tau & 0 & 0 & 0
\end{array}\right)
\left(\begin{array}{c}T_1\\T_2\\T_3\\\hline N_1\\N_2\\N_3\\\hline B_1\\B_2\\B_3\end{array}\right).$$
However note that it is more natural to consider the Frenet-Serret equation as an equation for triples for $3$-dimensional vectors, as this equation really is an ODE on $SO(3,\mathbb{R})$, which is exactly the symmetry/ ambiguity group associated to the first problem (up to translations in $\mathbb{R}^3$).
