Does the Riemann-Christoffel curvature determine the connection?

Part of the difficulty in providing an answer to your question is the fact that the expression "the integrability condition" is a somewhat vague notion, and it's used in slightly different senses in different contexts.

The usual, somewhat imprecise, sense is that, for a given system of PDE, its 'integrability condition' is the set of necessary and sufficient conditions that it have local solutions. Slightly more precise usage would be, given a PDE system that depends on some data (in your case, the equations (1), where the data is the curvature $R$), to find the necessary and sufficient conditions on the data that ensure that the PDE system have local solutions.

In this sense, the Bianchi identities (your equation (2)) are usually not the integrability condition for the system (1) but are only part of the integrability condition, and they have to be interpreted appropriately. The point is that, when $R$ is specified, equation (2) is an algebraic equation $\Gamma$ must satisfy if it is to satisfy the first order PDE system (1). [It may not be immediately apparent that (2) is an algebraic equation on $\Gamma$, but if you look at the definition of $\nabla$, you'll see that the left hand side of (2) only involves the unknown coefficients of $\Gamma$ linearly with no differentiation of them, and the right hand side involves the unknown $S$, which is computed linearly from the unknown $\Gamma$.] Thus, a necessary condition on the data $R$ that (1) have a solution is that there be at least one solution $\overline\Gamma$ to the inhomogeneous linear algebraic system (2) (whose coefficients come from $R$ and its first derivatives). Generally, this necessary condition is not sufficient, though, as examples in dimensions $4$ and above show.

However, what Deane Yang's reference eudml.org/doc/74779 (DeTurck & Talvacchia, 1987) shows is that, when (i) $n=3$, (ii) $R$ is sufficiently generic (in an appropriate sense) while satisfying the integrability condition (2), and (iii) $R$ is real-analytic, then the system (1) is locally solvable. The system of PDE for $\Gamma$ that has to be solved, though, is highly nonlinear and, in a sense, overdetermined, so that the Cartan-Kähler Theorem or one of its modern versions has to be applied. There is no uniqueness and no way explicitly to solve the equations for $\Gamma$ for a given generic $R$ that satisfies the integrability condition. In that sense, your second 'reverse question' has no good answer.

Finally, let me remark that one could conceivably want to answer the more restrictive problem of specifying both the curvature $R$ and torsion $S$ of a connection $\Gamma$ on a manifold $M$. This problem is overdetermined even when $n=3$ and, generally, has no solution. This problem also has an additional set of integrability conditions of the form $\nabla S = F(R,S)$ (for an explicit expression $F(R,S)$ that one can write down easily) and which are also inhomogeneous linear algebraic in $\Gamma$, in addition to the Bianchi conditions $\nabla R = G(R,S)$ that you already have written down as (2). In general, these combined integrability conditions are not sufficient for the generic pair $(S,R)$ that satisfy them in order for a solution $\Gamma$ to exist.