Is it obligatory to prioritize the repayment of debts or expiations over the sacrifice of the 'Aqiqah for a newborn, and is it permissible to use the newborn's gift (money) for the 'Aqiqah while there are debts or expiations outstanding?

If the debtor's debt is deferred and the creditor does not demand it, there is no harm in him slaughtering the aqiqah, as he is considered a benefactor as long as he is capable of repaying the debt. Some scholars have even permitted borrowing for the purpose of an udhiyah (sacrificial animal for Eid al-Adha) or aqiqah, provided one is capable of repayment, as stated by Sheikh al-Islam Ibn Taymiyyah and al-Buhuti. The Malikis even permit selling one's Friday garment and borrowing for the aqiqah.

As for prioritizing the aqiqah over expiations for oaths (kaffarat al-ayman) when it's not possible to combine them, this depends on whether the expiation for an oath is considered immediate (fawri) or flexible (mutarakhi). If it is deemed immediate, then the aqiqah should not be prioritized over the expiations. However, this does not mean the aqiqah is not accepted; rather, the reward for the aqiqah is obtained while the sin of delaying the expiation remains.