Is one obligated to take the initiative to pay off debts and expiations related to one's liability—whether oaths' expiations or others—using the gift given for the newborn, or is it permissible to offer the 'Aqiqah with it? And is the 'Aqiqah accepted if there are outstanding debts or expiations that have not yet been paid off?

Originally, paying off an immediate debt takes precedence over offering the 'aqiqah. As for a deferred debt, there is no harm in offering the 'aqiqah if doing so does not affect the timely payment of installments. Expelling the expiation for an oath (kaffarat al-yamin) is immediately obligatory according to the majority of scholars. So, when you receive money, you must feed ten poor persons or clothe them, and this takes precedence over the 'aqiqah if it is not possible to combine both. If a person offers the 'aqiqah before paying off an obligatory debt or an obligatory expiation, he is sinful for delaying an obligation, but his 'aqiqah is valid.