I think ionic compounds dissolve in just about any compound including nonpolar. My reasoning is this:
Ionic compounds don't always dissolve in just about any compound. Consider sodium chloride. This isn't soluble in acetone.
Neutral electric charge attracts both positive and negative(This I
know from doing things with static electricity and putting it on a
neutral surface).
If something has no net charge (i.e. is neutral) it won't be electrically attracting anything. Only oppositely charged things attract.
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).
Not all strong acids or bases are ionic. Consider sulfuric acid. The S-O and O-H bonds, granted, have some ionic character, but are generally characterized as covalent. Also consider the halide series of binary acids - HCl, HBr, HI. Strength increases in that order, yet ionic character decreases in that order.
Plus a base isn't necessarily ionic. NaOH is an ionic salt, and yes, it's a strongly basic salt. But the hydroxide ion is also a strong base. In isolation, it's not ionic.
Also you should define what a dimer is before incorrectly applying it. The definition of a dimer according to IUPAC is:
The transformation of a molecular entity $\ce{A}$ to give a molecular entity $\ce{A2}$.
So no, the sodium and chloride ions cannot dimerize in the fashion you suggested - with sodium ion coming together with chloride ion. That's not dimerization. Now, if a chlorine radical joined with another chlorine radical, that would be dimerization, since both starting molecular entities are the same.
Do ionic compounds such as NaCl dissolve in just about every solvent
from polar to nonpolar to liquid ionic?
No, and this has been illustrated by user137's excellent demonstration. And consider this table of solubility products. A bunch of ionic compounds, yes. But all of them are considered semi-soluble salts because of their low degree of dissociation into their constituent ions (hence the miniscule Ksp values).
