Is the marriage contracted between me and the second man Islamically permissible and valid, considering that my guardian was not present at the عقد (marriage contract) and my brother merely approved over the phone, and no مهر (dowry) was agreed upon? And is an عدة (waiting period) obligatory for me after this divorce? And am I sinful due to my ignorance of the correct conditions for marriage?

Marriage without a guardian (wali) is invalid according to the majority of jurists, and the consent of the youngest brother is not sufficient because he is not a guardian. According to the preponderant opinion, this contract is not legally valid.

However, according to the Madhhab of Abu Hanifa, the marriage is valid if the woman is an adult and of sound mind. The woman's intention to make herself lawful for her first husband does not affect its validity, as long as the husband married her out of genuine desire and was not stipulated to divorce her.

The absence of mentioning the dowry in the contract does not affect its validity (she is entitled to a dowry equivalent to that of women of her status), nor does the lack of publicizing the marriage affect the validity of the contract (publicity is recommended).

The woman must observe the 'iddah (waiting period) from this marriage when the cause for 'iddah arises. Does this marriage make her lawful for her first husband?

According to those who deem it valid: it makes her lawful. According to those who deem it invalid: it does not make her lawful; because a فاسد (corrupt/defective) marriage is not truly a marriage, and even if there is a difference of opinion regarding its invalidity and he consummated the marriage with her, she does not become lawful for the first husband according to those who deem it invalid.