Ionic compounds dissolve in nonpolar compounds? I think ionic compounds dissolve in just about any compound including nonpolar. My reasoning is this:

Neutral electric charge attracts both positive and negative (This I know from doing things with static electricity and putting it on a neutral surface).

Ionic compounds like strong acids and bases disassociate into individual ions, or if it is not dissolved but melted ionic dimers (like an individual Na+Cl- dimer for example).

Do ionic compounds such as NaCl dissolve in just about every solvent from polar to nonpolar to liquid ionic?
First, I think you have some misconceptions about electrical charge. There isn't neutral charge, it's either positive or negative, if it doesn't have either, it has no charge. Why are you seeing static charge interaction with neutral surfaces? It might be polarization, where a charged particle warps the electron cloud around a neutral atom generating partial charges around the neutral atom. Without knowing more about your experiment I can't say more.

Meanwhile I did an experiment of my own. I took 5 20mL scintillation vials, and added about 0.5g of NaCl to each. I then added 5mL of deionized water, methanol, 200 proof denatured ethanol, isopropanol, or acetone to each vial and tried to dissolve the salt. Only water dissolved all of the salt. I can't say there isn't ANY salt dissolved in the other solvents, but clearly the solubility is solvent dependent, and only water is polar enough to dissolve all the salt.