Is there any relation between spin and gravity? Is there any relation between quantum spin and gravity?
Are there experiments that show or motivate that there is or isn't a relation between the two?
 A: All things are related to each other in one way or another. So I will instead answer the following question:

How is spin related to gravity?

First, there are two different things people usually mean by spin. It can be the physical spin of elementary particles ($1/2$ for standard model fermions, $1$ for gauge bosons, $0$ for the Higgs and $2$ for the hypothetical graviton). But also, the labels of irreducible representations of groups are called "spins". Particle spins also label the representations of the (Lie algebra of the) rotation group in 3 spatial dimensions.
How the physical spin of elementary particles relates to gravity? Well, to be honest, nobody knows for sure. But here is what we do know already: in the presence of gravity, elementary particles (fermions for concreteness) couple to the tetrad field $e_{\mu}^I$ as follows:
$$ \mathcal{L}_f = \sqrt{|e|} \cdot \bar{\psi} \left( i \gamma^I e^{\mu}_I D_{\mu} - m \right) \psi. $$
Particle spins label the representations of the Lorentz group $SO(3,1)$ which acts on internal (capital latin) indices.
Note that this might be completely wrong in the realm of quantum gravity. After all, this is an effective Lagrangian coming from unknown fundamental degrees of freedom.
In the comments, several people mentioned mentioned spin networks and loop quantum gravity. Spin networks are hypothesized to label the quantum states of the background-independent quantum theory of gravity. They are combinatorial graphs with edges labeled by irreducible representations of a certain group (or quantum group). These representations are often loosely called spins, but they have nothing to do with spins of elementary particles. It's just bad terminology.
How are spin networks related to elementary particles and their spin? Well, this is also unclear at the moment. I see three possibilities:


*

*Not related at all. The spin network idea, though aesthetically pleasing, might turn out to be wrong.

*Spin networks have additional quantum numbers (labels on nodes and links) corresponding to another gauge group, from which the model of elementary particles emerges.

*Elementary particles are not fundamental, but emergent. They are born during the phase transition when the diffeomorphism group becomes spontaneously broken and spacetime emerges.
A: The involvement of spin in gravitation is an inescapable corollary of the marriage between gravity and quantum field theory (QFT). QFT (and for that matter the Standard Model) hinges on the existence of Dirac fermion and the rational way of incorporate gravity into Dirac spinor Lagrangian is the Einstein–Cartan–Sciama–Kibble theory, characterized by
$$
\mathcal{L}_f = \sqrt{|e|} \cdot \bar{\psi} \left( i \gamma^I e^{\mu}_I D_{\mu} - m \right) \psi.
$$
This Lagrangian is spelled out in @Solenodon Paradoxus' answer too. Here I would like to highlight that the Lorentz-covariant derivative 
$$
D_{\mu} = \partial_{\mu} + \omega_{\mu}^{JK}\gamma_J\gamma_K,
$$
involves the spin connection $\omega_{\mu}^{JK}$, which couples to spin and is essential in maintaining the local Lorentz-covariance of $D_{\mu}\psi$. 
In the absence of fermions and spin, the spin connection $\omega_{\mu}^{JK}$ can be expressed by the tetrad field $e^{\mu}_I$ pursuant to the zero torsion condition, which recovers the Einsein's version of gravity. 
A: This is potentially a very deep question. The answer from our current knowledge would probably have to be no. However, the spin of elementary particles is a very fundamental property. And as our knowledge of gravity increases (at the quantum level) I believe a clear connection between spin and gravity will emerge.
A: Whereas Newtonian gravity is based off of Mass only, the gravity of General Relativity has the Stress-Energy Tensor as its source. 
This tensor takes into account Mass, which has energy equivalence with $E=mc^2$, mechanical energy like rotation, or kinetic, and what might be generally called pressure or momentum flux, the change in a component of momentum as it varies by space. 
The Stress-Energy tensor takes into account all sources of energy and momentum, including rotation. Spin contributes to rotational energy, and so appears in the Stress-Energy tensor, thus contributing to gravity. 
Spin also gets carried along in spin-orbital coupling leading to a certain organization in a system that can induce some regularity in a system. 
The most significant contribution of spin in the gravity picture is where it is almost completely separate from gravity, but cancels it out in a sense. Neutron stars are stable because of Degeneracy Pressure. Fermions, like neutrons and electrons, particles with half-integer spin, cannot occupy the same quantum state according to the Pauli Exclusion Principle. Gravity tries to pull particles into the same state, Degeneracy Pressure tries to push them out of this state. Spin wins this battle in neutron stars and loses in the case of black holes. 
A: I imagine the spin (of the electron) as a small elementary knot in the spacetime; seen from far away, is a small gravitational wrinkle, corresponding to the electron mass.
