I understand from 1 that reducing the structure group $GL^+(4,\mathbb{R})$ of the frame field $FX$ of a world manifold $X^4$ to the $SO(4,\mathbb{R})$ group entails the Levi-Civita connection as the (unique torsion-free) connection that preserves the reduction.
Can we define the spin connection as a structure group reduction? For instance does a structure reduction from $GL^+(4,\mathbb{R})$ or $SL(4,\mathbb{R})$ to either $Spin(4)$ or $Spin^c(4)$ produces the spin connection?
No reductions of structure group are not sufficient. The groups $Spin(4)$ and $Spin^c(4)$ are not realized as subgroups of $GL^+(4,\mathbb R)$. The natural homomorphism $Spin(4)\to GL^+(4,\mathbb R)$ (coming from the action of $Spin(4)$ on $\mathbb R^4$) has image $SO(4)$ and two element kernel. To get to the spin connection you first have to reduce the structure group to $SO(4)$. In a second step, you have to extend the structure group to $Spin(4)$. Basically, this means that you have to define a principal $Spin(4)$-bundle such that the reduction by the two element kernel of the canonical homomorphism $Spin(4)\to SO(4)$ reproduces the $SO(4)$-reduction of the frame bundle you started from.
Such an extension exists only if the second Stiefel-Whitney class of your manifold $M$ (a characteristic class in $H^2(M,\mathbb Z_2)$) is zero. If this is the case, then the extension is not unique in general, this is related to the first Stiefel-Whitney class in $H^1(M,\mathbb Z_2)$. Therefore the whole process is referred to as "a choice of spin-structure". For $Spin^c(4)$ the story is similar, apart from the fact that $Spin^c(4)$-structures always exist. (This is not true in higher dimensions.)