Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
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.
Should a person feel pleased, or have a virtuous vision after offering Istikhara (guidance prayer) in order to do what he/she had offered it for?
The result of the Istikhaarah is not necessarily that a person sees something (in his dream), or feels pleased, but the most important result of the Istikhaarah is whether a person is enabled to do a given matter or not.
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.