Is hair extension permissible?

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

It is impermissible for the man and the woman to apply a human`s hair as extension, but using artificial hair is permissible for the married woman after obtaining the approval of her husband, and provided that non-Mahrams (Marriageable men) don`t see her. And Allah Knows Best.

What is the ruling on shaving the head of the newborn?

It is desirable to shave the head of the newborn on the seventh day from delivery, and the weight of his/her hair is given in gold , or silver as charity.

Is it permissible for a woman who broke fasting due to delivery to pay a ransom ?

She should make up for the missed fasting days once she becomes ritually pure, but it is impermissible for her to pay a ransom while being able to fast.

Is Zakah(obligatory charity) due on rented property, and how is it calculated?

Zakah isn`t due on the rented property itself, but on the saved income from that property if it reached Nissab(minimum amount liable for Zakah), and a whole lunar year had lapsed over saving it.