Does determining the mahr constitute a marriage, and is the girl considered a wife who can be seen without a hijab and be secluded with?

The engagement is a promise of marriage and a prelude to it. It does not require specific wordings. Reciting Al-Fatihah at the time of engagement has no basis in Islamic law. Mentioning and specifying the dowry does not give the engagement the ruling of a marriage contract. The marriage contract is concluded by offer and acceptance between the woman's guardian or his proxy, and the husband's acceptance or his proxy, in the presence of two witnesses. If the pillars of the contract are not met, the fiancée is not a wife to the suitor; rather, she remains a stranger to him, and she is not lawful for him except through a valid Islamic marriage contract.