Edit: the OP seems to be interested in differentiating a composite object, i.e. a field, say $\phi$, which has different types of indices. For example, let us say that $\phi^{\nu A j}$, where $\nu$ is a spacetime index, $A$ is a Dirac spinor index ($A = 1, \dots, 4$) and $j = 1, 2$ is an index corresponding to the trivial bundle $M \times \mathbb{C}^2$, corresponding to the gauge group $G = SU(2)$, where $M$ denotes our spacetime manifold, with metric $g_{\mu \nu}$. Assume that we have a gauge connection $A_\mu$ on the principal $SU(2)$ bundle $M \times SU(2)$. Then

$$D_\mu \phi^{\nu A i} = \partial_\mu \phi^{\nu A i} + \Gamma_{\mu \rho}^\nu \phi^{\rho A i} + \omega_{\mu B}^A \phi^{\nu B i} + i A_{\mu k}^j \phi^{\mu A k},$$ or something close to that, where $\Gamma$ is the Levi-Civita connection, and $\omega$ is the spin connection (please note that there may be factors missing and this is just to give you an idea of how to differentiate a "mixed" type of object, so to speak).

Edit: the OP seems to be interested in differentiating a composite object, i.e. a field, say $\phi$, which has different types of indices. For example, let us say that $\phi^{\nu j}$, where $\nu$ is a spacetime index and $j = 1, 2$ is an index corresponding to the trivial bundle $M \times \mathbb{C}^2$, corresponding to the gauge group $G = SU(2)$, where $M$ denotes our spacetime manifold, with metric $g_{\mu \nu}$. Assume that we have a gauge connection $A_\mu$ on the principal $SU(2)$ bundle $M \times SU(2)$. Note that, for each $\mu$, $A_\mu$ is an element in the Lie algebra of $SU(2)$ (more precisely in a bundle of Lie algebras of $SU(2)$, called the adjoint bundle, but this is a technicality, which we can omit for the time being). So, for each $\mu$, $A_{\mu}$ is a complex $2$ by $2$ skew-hermitian matrix for mathematicians, though for physicists $A_\mu$ would be a complex $2$ by $2$ hermitian matrix, but physicists would then write $i A_{\mu}$, so the two notations are equivalent, because $i$ times a hermitian matrix is a skew-hermitian matrix. But my point is, for each $\mu$, $A_\mu$ is a $2$ by $2$ matrix, and so it has two extra indices, say $j$ and $k$, with each of them having only two possible values, say $1$ and $2$.

With these remarks out of the way, differentiating the field $\phi$ using the covariant derivative $D$ gives the following.

$$D_\mu \phi^{\nu j} = \partial_\mu \phi^{\nu j} + \omega_{\mu ab} (\sigma^{ab})^\nu_{\phantom{\nu} \rho} \phi^{\rho j} + i (A_\mu)^j_{\phantom{j}k} \phi^{\nu k}.$$

I tried to use a notation close to the OP's notation, but then again, I don't know the author's conventions, so there may be silly factors missing from what I wrote. I only wrote it to hopefully help with the OP's understanding.

For the first part of your post, related to tetrads, a tetrad is basically a smooth locally defined orthonormal coframe with respect to $g_{\mu \nu}$. More precisely, it consists of $4$ (assuming the dimension of spacetime is $4$) locally defined $1$-forms, $e^1_\mu, \dots, e^4_\mu$ which are orthonormal with respect to $g_{\mu \nu}$. Taking into account that the metric has Lorentzian signature, this precisely amounts to the first equation you wrote down, defining a tetrad.

You can also think of a tetrad $e^a_\mu$ at a point $p$ as defining a linear isometry map from $T_p$, with the metric $g_{\mu \nu}$, to $T_p$, with the Minkowski metric $\eta_{ab}$.

When you differentiate a tetrad, you need to use the Levi-Civita connection of $g_{\mu \nu}$ for the $g$ index of $e$ (namely the lower index of the tetrad) and you have a minus sign because you are differentiating a $1$-form, and you need to use the Levi-Civita connection of the Minkowski $\eta_{ab}$ metric to differentiate the $\eta$ index (the upper index of the tetrad). Note that if we were using local coordinates, then $\omega$ would be $0$ but, because we are using a tetrad instead, $\omega$ is in general non-zero.

For the second part of your post, to differentiate the $\gamma$s, note that each $\gamma$ is a linear map from $T_p$ to $\operatorname{End}(\mathscr{S}_p)$, the space of linear endomorphisms of the spinor bundle $\mathscr{S}_p$ at $p$. Thus each $\gamma$ has a "vector" type of index and two "spinor" type of indices. To differentiate the "vector" type index you use the Levi-Civita connection of $g_{\mu \nu}$, and to differentiate the two "spinor" type indices you use the spin connection.

As you can see, in both these cases we did not use the gauge connection. So, when do we use the gauge connection? A gauge connection is a connection on principal fiber bundle $E$ on spacetime. You use it to differentiate sections of "associated" bundles of $E$.

For the first part of your post, related to tetrads, a tetrad is basically a smooth locally defined orthonormal coframe with respect to $g_{\mu \nu}$. More precisely, it consists of $4$ (assuming the dimension of spacetime is $4$) locally defined $1$-forms, $e^1_\mu, \dots, e^4_\mu$ which are orthonormal with respect to $g_{\mu \nu}$. Taking into account that the metric has Lorentzian signature, this precisely amounts to the first equation you wrote down, defining a tetrad.

You can also think of a tetrad $e^a_\mu$ at a point $p$ as defining a linear isometry map from $T_p$, with the metric $g_{\mu \nu}$, to $T_p$, with the Minkowski metric $\eta_{ab}$.

When you differentiate a tetrad, you need to use the Levi-Civita connection of $g_{\mu \nu}$ for the $g$ index of $e$ (namely the lower index of the tetrad) and you have a minus sign because you are differentiating a $1$-form, and you need to use the Levi-Civita connection of the Minkowski $\eta_{ab}$ metric to differentiate the $\eta$ index (the upper index of the tetrad). Note that if we were using local coordinates, then $\omega$ would be $0$ but, because we are using a tetrad instead, $\omega$ is in general non-zero.

For the second part of your post, to differentiate the $\gamma$s, note that each $\gamma$ is a linear map from $T_p$ to $\operatorname{End}(\mathscr{S}_p)$, the space of linear endomorphisms of the spinor bundle $\mathscr{S}_p$ at $p$. Thus each $\gamma$ has a "vector" type of index and two "spinor" type of indices. To differentiate the "vector" type index you use the Levi-Civita connection of $g_{\mu \nu}$, and to differentiate the two "spinor" type indices you use the spin connection.

As you can see, in both these cases we did not use the gauge connection. So, when do we use the gauge connection? A gauge connection is a connection on principal fiber bundle $E$ on spacetime. You use it to differentiate sections of "associated" bundles of $E$.

Edit: the OP seems to be interested in differentiating a composite object, i.e. a field, say $\phi$, which has different types of indices. For example, let us say that $\phi^{\nu A j}$, where $\nu$ is a spacetime index, $A$ is a Dirac spinor index ($A = 1, \dots, 4$) and $j = 1, 2$ is an index corresponding to the trivial bundle $M \times \mathbb{C}^2$, corresponding to the gauge group $G = SU(2)$, where $M$ denotes our spacetime manifold, with metric $g_{\mu \nu}$. Assume that we have a gauge connection $A_\mu$ on the principal $SU(2)$ bundle $M \times SU(2)$. Then

$$D_\mu \phi^{\nu A i} = \partial_\mu \phi^{\nu A i} + \Gamma_{\mu \rho}^\nu \phi^{\rho A i} + \omega_{\mu B}^A \phi^{\nu B i} + i A_{\mu k}^j \phi^{\mu A k},$$ or something close to that, where $\Gamma$ is the Levi-Civita connection, and $\omega$ is the spin connection (please note that there may be factors missing and this is just to give you an idea of how to differentiate a "mixed" type of object, so to speak).