Is it permissible to divorce a husband who lives in another country, suffers from psychological problems, behaves inappropriately, and calls his wife by indecent names?

If a marriage "on paper" fulfills the Shariah requirements (wali, two just witnesses, mahr), then it is valid, and official documentation is not a condition for its validity. The husband's abusive and insulting words permit a request for divorce, provided the harm is proven by two just witnesses, and the matter can be taken to a Shariah court. However, if the words do not amount to abuse and insults, they are not considered a valid reason for divorce.