Is it valid for the 'aqīqah to be performed using the newborn child's own wealth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for the guardian to perform the 'aqīqah using the newborn child's own wealth, because the 'aqīqah is a voluntary act of giving (tabarru'), and a guardian is prohibited from making voluntary expenditures from the child's wealth. Should he do so, he becomes financially liable for what he spent. And Allah Almighty knows best.
What is the ruling on delaying Zakat al-Fitr until after Eid day?
It is forbidden to delay Zakat al-Fitr beyond the sunset of Eid day. If someone delays it past Eid day without a valid excuse, they have committed a sin and must immediately make up for it because their obligation remains unfulfilled, and they must clear their responsibility.
What do you say about someone who stays up all night on Laylat al-Qadr but misses Fajr prayer?
Such a person has deprived themselves of a great reward. Performing obligatory acts is more beloved to Allah than voluntary acts. Fajr prayer is an obligation, and performing it in congregation is equivalent to worshiping for the entire night. So how can someone neglect it while striving for voluntary prayers?!
What is the ruling on ablution if vaginal discharge is expelled, and is it impure?
If these discharges exit from the external genitalia (apparent part of the vagina), they are not impure and do not invalidate ablution. If they exit from the internal part, they are impure and do invalidate ablution. If it is uncertain whether they are from the internal or external part, they are not impure and do not invalidate ablution.
The apparent part is what becomes visible when sitting, and what the husband's penis reaches during intercourse is considered part of the apparent. The internal part is what is beyond that. And Allah the Almighty knows best.