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.
Which is better during the day in Ramadan: reciting the Quran or performing voluntary prayers?
A Muslim should establish a daily schedule for reciting the Quran in Ramadan.
Once they have completed their daily portion (known as a wird or hizb), they can engage in other acts of worship, including voluntary prayers (nawafil). Prayer itself is called "Quran", because a significant part of it involves reciting the Quran.
Allah says: "BE CONSTANT in [thy] prayer from the time when the sun has passed its zenith till the darkness of night, and [be ever mindful of its] recitation at dawn: for, behold, the recitation [of prayer] at dawn is indeed witnessed [by all that is holy]." [Al-Isra`/78]
Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
Is the marriage, which lacks a valid legal contract, a guardian, and a court registration, valid?
It is incumbent that a valid marriage contract be concluded in the presence of a guardian and two trustful witnesses, and it should be registered in the court to protect the rights of the wife. Actually, a valid marriage contract is what differentiates between sound marriage and fornication.