“I also have heard in the name of the Chazon Ish (I read it in Oz Vehadar levusha by R falk in the name of the CH”I as well) that a woman appearing in public in a state run according to Torah law would be stoned.”
This makes no sense, and must be a mistake. Even Rambam, who is the most restrictive on the issue of women outside the home, says women can leave the home once a month (and this must have been affected by his social context of being in a medieval anti-Semitic Muslim country).