Is a marriage conducted in another Arab country (such as Syria), with legal recognition, witnesses, a dowry (Mahr), and a deferred dowry (Mu’akhkhar), considered religiously valid (Shar’i) and permissible (Halal), given that the father of the girl refused her marriage to the suitor for flimsy reasons related to his origin and income, and because he is not qualified for guardianship over her since he does not pray and drinks alcohol, and knowing the potential social consequences?

Your online relationship with this girl is forbidden, and you must repent and cut off the relationship immediately. If the girl's father insults Allah and His Messenger, he is an apostate and has no guardianship over his daughter in marriage, but the ruling of apostasy is for the judiciary. For the girl to be rid of her father's 'adhl (unjust prevention of marriage), she must resort to the judiciary and authorities for protection. She can also present her case to scholars or callers to Islam. If her fear persists, she must leave her father's house and live with her relatives. As for your traveling to another country to get married, it is not permissible due to her traveling without a mahram and the impermissibility of a woman marrying without a guardian. The judge is the one who marries her in case of 'adhl by the guardian.