What is the ruling on a woman driving a car?

All Perfect Praise is to Allah, The Lord of The Worlds, and may His Peace and Blessinsg be upn our Prophet Muhammad and upon all of his family and compnions.

It is permissible for the woman to drive a car, but it is impermissible for her to travel unaccompanied by a Mahram (Unmarriageable kin). And Allah The Exalted Knows Best.

Zakah due on a woman`s jewelry?

Obligatory Zakah (obligatory charity) isn`t due on a woman`s jewelry used for adornment, provided that the value of each piece is within the range amongst common market price. And Allah Knows Best.

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.

Is buying shared land from someone who has taken possession of it considered halal or haram?

The shared land is possessed by the country i.e. for all people; therefore, it shouldn't be possessed/occupied save by it's guardian/owner permission,thusly it can't be possessed save by having owners' permission. Finally, buying it from a person who don't possess it is impermissible. And Allah Knows Best.