What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
Is it permissible for the mother to give her children from the Zakah (obligatory charity) of her money?
It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.
Is it recommended (mustaḥabb) for the one offering a voluntary sacrifice (uḍḥiyyat taṭawwuʿ) to eat from its meat?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is highly recommended (Mustahabb) for the person offering a voluntary Udhiyah (sacrificial animal) to eat from its meat, but it is not an obligatory requirement (Wajib). Allah the Almighty says {what means}: "So eat of them and feed the needy who do not beg and the beggar. Thus have We subjected them to you that you may be grateful." (Al-Hajj/ 36)
"So eat of them": This directive is an encouraging recommendation, not a binding command.
The Qani’ (the needy who do not beg): Refers to a poor person who refrains from asking people for help out of dignity.
The Mu’tarr (the beggar): Refers to a poor person who explicitly asks others for assistance. And Allah the Almighty Knows Best.
Is it permissible to give multiple mudds (portions) of fidyah to a single person?
Yes, it is permissible to give multiple mudd of fidyah to a single person, because each day of fasting is considered an independent act of worship.