Eric,
1) if you want high concentration of electrons, you can have a few dozens of them in a large atom - does increasing Z make LENR more probable?
These electrons effectively screen the charge of nucleus down to zero ... but only asymptotically (and electric dipole/qudrupole/octupole may remain). While getting really close, this screening drops down to zero ( https://en.wikipedia.org/wiki/Shell_theorem ) - making essential only the single electron which remains between the two nuclei.
Anions can have stronger screening, but again only asymptotically - it doesn't help when the two nuclei are really close.
Electron concentration might be helpful in some initial stage, but the most of assistance is required in the final state - when the nuclei approach ~10^-15m distance so that nuclear force could take from here - in such distances there is just no place for a second electron due to tiny mass and huge Coulomb force.
2) Applying external magnetic or electric field perturbs the dynamical equilibrium (Zeeman, Stark). Classically, the electron orbits became a bit shifted, and Gryzinski claims nearly perfect agreement for such calculations of diamagnetic properties ( http://www.sciencedirect.com/s…icle/pii/0304885387903337 for He, Ne, Ar, NaCl, KCl and CaCl2).
Anyway, we are talking here about eV-scale of changes - very far from the required for nuclear transitions.
3,4) I see you are talking about more general LENR, while I was thinking about the ones finally leading to excess energy - not from radioactive isotopes or fission.
Neglecting situations with 782keVs for going through neutron, such excess energy requires crossing the Coulomb barrier - what, if true, requires electron staying between the nuclei for a sufficient time.
Electron concentration, high energy electrons, external field applied, van der Waals force etc. might have some influence on the initial state.
But the energy of Coulomb barrier is 1/r: crossing the last femtometers require more energy than getting to 10fm distance - this final state is the most crucial, and electron between the two nuclei seems the only factor which could really help.