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.
What is the ruling of Islamic Law on the prayer of zawal?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is recommended (sunnah) to pray four rakʿāt — either with a single tasleem or as two separate sets of two rakʿāt — immediately following the sun's decline from its zenith (zawāl). This prayer is distinct from the regular Sunnah prayer of Ẓuhr (sunnat al-ẓuhr al-rātibah), as explicitly stated by the Shāfiʿī jurists.
It is mentioned in Nihāyat al-Muḥtāj: "The prayer of zawāl is offered after the sun's decline — so were one to perform it before that, it would not count. It consists of two or four rakʿāt and is distinct from the Sunnah of Ẓuhr, as is evident from the fact that it is mentioned separately after the regular Sunnah prayers, and it becomes a make-up prayer (qaḍāʾ) if a long period of time passes by customary reckoning... Al-ʿAlqamī stated: 'Scholars refer to this as the Sunnah of Zawāl, and it is distinct from the four rakʿāt that constitute the Sunnah of Ẓuhr.' Our shaykh said: Al-Ḥāfiẓ al-ʿIrāqī stated that among those who explicitly affirmed its recommendation was al-Ghazālī in al-Iḥyāʾ, in the chapter on devotional litanies, noting that there is no tasleem between them — meaning there is no break between each pair of rakʿāt."
The time of the sun's decline (zawāl) marks the very beginning of the time for the Ẓuhr prayer.
And Allah the Almighty knows best.
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 swallowing phlegm while fasting?
If a fasting person intentionally swallows phlegm, their fast is invalid, as it is possible to avoid it.
However, if they swallow it unintentionally, their fast remains valid.
Phlegm is impure and should be avoided.