Do Cooper pairs act like Cheshire cats? Could the pairing up of electrons be explained by their spin being in a different position? 
What would separating the spin from an electron in matter do in theory?
 A: Yes, pairing electrons can be spatially separated. In the BCS theory the coherence length $\xi$, also known as the size of the Cooper pair, is the spatial extent over which electrons are expected to pair up. It depends on the material and is given by $$\xi \propto v_{F}/\Delta$$ where $v_{F}$ and $\Delta$ are the Fermi velocity in the normal state and the superconducting gap respectively. For some metals this can be in the microns range, i.e. $\approx 10^4$ times the lattice constant. As a matter of fact, BCS Cooper pairs are not “hard core” like a diatomic molecule. Since they extend over such a large distance, copper pairs “overlap.” This can be pictured as the right side of:
                                       
The left side shows the “hard core” paired fermions. One can normally tune between BEC and BCS limits in cold atom experiments.
As for your second question, I think you might be referring to the phenomenon called “spin-charge separation.” The notion of this new phenomenon was studied mostly in the content of high-$T_{c}$ cuprate superconductors. The parent state (i.e. undoped state) of the cuprates form an antiferromagnetic Mott insulator (see figure below).
                                               
In this material, one electron (or hole) is localized to one copper site due to Coulomb repulsion. However, due to the so-called “virtual hopping” of electrons from/to a copper site (blue circle) to/from the oxygen site (pink circle), the spins on nearest neighbor copper sites have opposite spin (see figure above). This is antiferromagnetic configuration is the ground state of the system. A spin flip would raise the energy of the system (i.e. excited state). So instead of exciting electron-hole pairs, like (say) a semiconductor, which carry both spin and charge, excitations in the antiferromagnetic system carry only spin; charge is localized to each copper site due to Coulomb repulsion.
Some theories suggest that pairing in cuprate superconductors is mediated by spin fluctuations. A cartoon of this is shown in:
                      
An electron traveling with a spin up (left) ferromagnetically polarizes the spin medium around it and hence pairs with another traveling spin (right) or vice-versa. According to this theory, this spin fluctuation causes the formation of Cooper pairs instead of electron-phonon interaction, like in BCS.
