Is it permissible for a wife to perform Salāt al-Istikhārah concerning obligatory matters or Sharīʿah rulings, and what is the Sharʿī ruling in this case, given that the husband wishes to mend his home and does not desire divorce, while the wife did not feel at ease after Salāt al-Istikhārah regarding the continuation of the marriage?

You erred in speaking to a foreign woman, but your repentance is accepted. Your wife erred in her defiance (nushuz) and her baseless request for divorce. She must repent and obey you, because your previous disobedience does not permit her to request a divorce, and you have repented from it. It is not permissible for her to seek a divorce without cause, nor to perform Istikhara (prayer for guidance) for something that is impermissible. She must fear Allah and return to her home obediently. If she insists on separation, then she may seek a khul' (divorce by mutual consent) by returning the dowry or whatever you both agree upon.