Can someone who begins a voluntary fast break it?
It is preferable for someone who begins an act of worship not to break it.
Allah the Exalted has said {what means}: "and let not your [good] deeds come to nought!" [Muhammad/33].
However, if a person starts a voluntary fast (nafl) and needs to break it, they are going against what is preferable, but there is no sin upon them.
What is the Islamic ruling on the Udhiyah (sacrificial offfering)?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The Udhiyah (sacrificial offering) is a Confirmed Sunnah (Sunnah Mu’akkadah) for every adult Muslim of sound mind who possesses the financial means, whether they are a resident, a traveler, or a pilgrim (Haj). This is based on the statement of the Prophet ﷺ: 'When the ten days [of Dhu al-Hijjah] begin and one of you desires to offer a sacrifice, let him not touch [cut] anything of his hair or skin' [Narrated by Muslim].
The point of evidence (Wajh al-Dalalah) here is that the Prophet ﷺ linked the sacrifice to the individual's will and desire by saying, 'and one of you desires.' This indicates that it is not obligatory (Wajib); had it been mandatory, he would have simply said, 'let him not touch his hair until he sacrifices' [without making it conditional upon desire].
Furthermore, it is narrated that Abu Bakr and Umar (may Allah be pleased with them both) would sometimes refrain from offering the sacrifice out of fear that people might mistakenly view it as an obligatory duty [Narrated by al-Bayhaqi and others with a good (Hasan) chain of transmission]. And Allah the Exalted knows best.
Does buying meat equivalent to the weight of a slaughtered animal and distributing it avail for an udhiyah?
Praise be to Allah, and prayers and peace be upon our Master the Messenger of Allah.
That does not avail for the sacrifice (uḍḥiyyah), because the sacrifice necessarily requires slaughtering an animal from the category of anʿām (camels, cattle, and sheep/goats) within a specified time.
Rather, that is considered a form of charity (ṣadaqah) for which one is rewarded, but not an udhiyah. And Allah Almighty 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.