Is valid seclusion (al-khalwa al-shar’iyya) without penetration considered equivalent to consummation (al-dukhool), thereby rendering a first divorce a minor irrevocable divorce (baynunah sughra) that necessitates a new contract and a new mahr, and consequently, the subsequent oaths (al-ayman al-muta’akhkhira) do not take effect due to the absence of an existing marriage? Or is seclusion considered actual consummation, and the subsequent oaths are counted as a major irrevocable divorce (baynunah kubra) that prohibits the wife from remarrying her former husband unless she first marries another man (muhallil)?

A valid seclusion is equivalent to consummation in establishing the mahr (dowry) and iddah (waiting period). This is the position of the Hanafis, Malikis, and Hanbalis. Al-Tahawi narrated the consensus of the Companions that for whoever closes a door and lowers a curtain, she is entitled to the mahr and is obliged to observe the iddah. Based on this, your first husband's divorce was valid, and his taking you back without a new mahr or contract was valid. Your second and third divorces are effective, and you are irrevocably divorced from him with a major divorce, meaning you are not permissible for him unless you marry another husband. A follower (muqallid) must adhere to the fatwa after acting upon it, by scholarly consensus.