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@KhaledT.: I didn't assume anything about a metric. This argument works for any affine connection. The choice of local coframing $\omega$ is for convenience (to avoid having to define and work on the full coframe bundle); it doesn't matter which coframing $\omega$ (holonomic or not holonomic) that one chooses.
Then you need to edit your question. You wrote, "I want to characterize which Riemannian manifolds $(M,h)$ [$\ldots$] admit solutions to the previous system.", which, to me, indicates that you are regarding the metric $h$ as a given that constrains the solutions. Perhaps you meant to write something like, "I want to understand the metrics $h=\sum e^k\otimes e^k$ defined by coframings $e$ that satisfy the above conditions".
Your phrase 'with $i_0$ and $j_0$ fixed' seems ambiguous to me. Do you mean 'for some given pair $i_0$ and $j_0$' or do you mean 'for any given pair $i_0$ and $j_0$'? In any case, it's not hard to give examples of functions $\Theta^i_{jk}$ for which there is no solution to the first system at all, even locally.
Can you say what you mean by $\alpha$ and $\beta$ being `arbitrary smooth functions'? Are they part of the unknowns or are they specified in advance? Note that in the case $\alpha \equiv\beta\equiv 0$, the function $\tau$ would be a constant $c$, and $\gamma$ would just have to satisfy $\gamma_{uv} = c^2\,\sin\gamma$. However, this is very exceptional. On an open set in $uv$-space where $\alpha\beta$ is nonvanishing, there is only a finite dimensional space of possible nontrivial solutions.
@Bumblebee: The 'missing information' in $\Psi$ is the mean curvature $H=\tfrac12(h_{11}+h_{22})$ and a choice of orientation of the surface. However, given $I$ and $\Psi$, one can recover the mean curvature $H$ up to a sign since, by the definition of $\Psi$ and the Gauss equation $H^2 = K - \det_I(\Psi)$, and the orientation of the surface can also be recovered up to a sign. Thus, there are analogs of the Gauss-Codazzi equations for the pair $(I,\Psi)$, but they are more algebraically complicated than for the pair $(I,I\!I)$.
I think you'll be able to answer this question yourself once you have the correct definitions. The actual definition is $\mathbb{C}\otimes T = T^{(1,0)}_a\oplus T^{(0,1)}_a$, where $T^{(1,0)}_a = \{ v - i\,av\ |\ v\in T\}$ and $T^{(0,1)}_a = \{ v + i\,av\ |\ v\in T\}$.
