As far as the conditions of marriage, only virgin girl requires the approval of her father or paternal Durand father for her marriage. Man does not need approval from his father or mother in his marriage, but it is very important to keep the respect of your parents and avoid hurting their feelings. You as a man are entitled to decide your marriage, without causing any harm or disrespect to your parents. You should try your best to convince them and get their blessings.
No unless you have concrete evidence that he is committing Haraam which can destroy the life of the girl, then you can give them the evidence and leave them to verify and decide.
Many people promise before marriage to embrace Islam or Shia Islam after marriage, just to get married with whom they want to marry, but they fail to fulfill the promise after marriage especially after passing years after marriage or after matrimonial misunderstands which are usual between couples. You must be sure the non Shia girl who wants to be married to you is really ready to follow Ahlul Bayt (AS) not for the sake of marriage but for the sake of Allah to a follower of Islam of the Prophet (SAWA) and Ahlul Bayt (AS).
Your parents might have this concern and don't wish to see you in trouble in the future.
'If this girl is ready to follow Ahlul Bayt (AS) for sake of the Truth, she should start following Ahlul Bayt (AS) now and not wait. That should indicate her intentions and might help your parents to believe that she is a sincere person in her promise.
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