Is it permissible for a woman to adorn herself with kohl (eyeliner) and go out wearing it?

Kohl is a form of adornment, and it is forbidden for a woman to display her adornment to anyone other than her husband or a Mahram. And Allah Knows Best.

Is it incumbent on a family to provide for its old handicapped son and take care of him?

The family and relatives - e.g. brothers - of such a person are obliged to spend on him and take care of him if he didn`t have money of his own.

Is wearing a cap permissible for boys, or girls?

All Perfect Praise is due to Allah, The Lord of The Worlds.

In principle, anything is permissible sol long as there is no Sharia evidence prohibiting it, thus wearing the cap by boys is permissible as long as it isn`t done in imitation of the disbelievers. Moreover, a Muslim woman should avoid all that which attracts the attention of men because whatever leads to the forbidden is forbidden. And Allah Knows Best.

What are the benefits of slaughtering an Aqeeqah?

All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.                                                                                                                                                                                                                          Extending thanks to Allah for His grace, expressing happiness for having a newborn, declaring lineage, and feeding the mother to compensate for the blood that she had lost during delivery.