Is it permissible to offer mandatory prayer while sitting?

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.                                                                                                                                                                                                                            Standing up is a pillar in the mandatory prayer and the prayer is not valid without it unless one cannot do it. As for Nafila (supererogatory prayer), one can perform it in the state of sitting even if he/she can stand, but his/her reward is half the reward of the person who prays while standing if he/sh has no legal excuse for such an act. And Allah Knows Best.

Is it permissible for a Muslim woman to leave her house adorned and without wearing her proper Islamic attire?

A Muslim woman isn't permitted to leave her house adorned and without wearing proper Islamic attire. And Allah Knows Best.

When Iddah is Initiated for Deceased's Wife?

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.

Iddah is observed on the same date at which her husband passed away for four months, but as for the pregnant, it is observed at the date she gives birth. And Allah Knows Best. 

I am an author, and I have books stored with me. Is it permissible to pay my zakat in the form of books containing beneficial knowledge for poor students, such as chemistry and mathematics, or religious books that spread Islamic sciences?

Zakat on wealth must be paid in cash from the wealth itself. Therefore, you should pay your zakat in monetary form so that the poor can purchase what they need. And Allah Knows Best.