What salts are/are not birefringent?
Yes, the only crystal system that is guaranteed not to be birefringent is cubic, because all of the other crystal systems have at least one axis distinct from the others. What matters is the symmetry not the centring, so this is equally true for face-centred or body-centred as for primitive crystals.

Sadly, however, there is no way to determine the crystal structure, or even the crystal system, simply from consideration of the formula - indeed, crystal structure prediction and prediction of polymorphism are active research fields. Counterexamples to your first generalisation include ammonium chloride, bromide, and iodide; to your second include ammonium fluoride and silver iodide.