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 doubting whether one or two prostrations were performed?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
If a worshiper is in doubt regarding the number of units (rak'ahs) or prostrations (sajdahs) performed, he must build upon the minimum (i.e., assume the lower number) and perform the prostration of forgetfulness (Sujud al-Sahw) before the Salam at the end of the prayer. This is based on the report from ‘Ata’ ibn Yasar that the Messenger of Allah ﷺ said: 'When anyone of you is in doubt about his Salat (prayer) and does not know how many he has prayed, three or four (Rak'at) he should cast aside his doubt and base his prayer on what he is sure of. Then, he should perform two prostrations before Taslim (salutation). If he has prayed five Rak'at, they will make his Salat (prayer) an even number for him and if he has prayed exactly four, they (i.e. two prostrations) will be humiliation for the devil..' (Narrated by Abu Dawud).
It is stated in Al-Muqaddimah al-Hadramiyyah: 'If one doubts [whether he performed] a bowing (ruku’), a prostration, or a rak'ah, he must perform it and prostrate [for forgetfulness], even if the doubt is removed before the Salam—unless the doubt is removed before he performs what would potentially be an addition. Thus, if he doubts whether he prayed three or four, he is obligated to build upon the minimum.' And Allah the Exalted knows best.
Is it permissible for someone with a physically demanding job, such as a baker or construction worker, to break their fast?
It is not permissible for someone with a physically demanding job to start the day intending to break their fast. They must make the intention to fast at night and begin fasting. However, if they reach a point where fasting becomes unbearably difficult, they may break their fast and make up for it later.
Does passing wind, from the anus without a smell, invalidate prayer?
If a person is certain of having passed wind even if it was with no smell, then both his/her ablution and prayer are invalidated, thus he/she should remake ablution and re-perform prayer.