Is it permissible for my divorced mother to offer a sacrifice from her social security allowance designated for divorced women, and is it permissible for her to appoint her daughter's husband to purchase and slaughter the sacrificial animal? And is my younger brother, who will slaughter our father's sacrifice, required to refrain from cutting his nails and hair, and what should he intend when slaughtering?

The Udhiyah (sacrifice) is a confirmed Sunnah for those who can afford it without undue hardship, and it is disliked for one who is capable to neglect it. If your mother is well-off, she should not neglect the Udhiyah. It is permissible for her to perform the Udhiyah with money from the government, as long as she has taken possession of it. It is also permissible for her to appoint her daughter's husband as her agent for purchasing and slaughtering. Similarly, it is permissible for your father to appoint your younger brother as his agent. As for the agent refraining from cutting hair and the like, the preponderant opinion is that the one addressed by this directive is the sacrificer himself, not the agent.