Is it permissible for a woman who has become pure before the end of forty days of confinement to perform different acts of worship? and is it lawful for her husband to have sex with her?

Once postpartum bleeding ceases and she is certain that it won`t happen again, she becomes pure, and so she should perform Ghusl (ritual bath), pray, and fast. She can also have sex with her husband even before the end of (40) days because the minimum of postpartum bleeding is a moment, and the maximum is (60) days. However, the postpartum bleeding for the majority of women is (40) days, but this does not apply to rest of them.

What is the ruling on reciting the Chapter after Al-Faatihah , and what should a worshiper who forgets it do?

All perfect praise be to Allah, The Lord of The Worlds.                                                                                                                                                                  Any other part of Qur'an that a worshiper reads after Al-Faatihah during the first and the second Rak`ahs is a Sunnah, and forgetting to do so doesn`t nullify prayer, and doesn`t require performing Sujood As-Sahw (prostration of forgetfulness). And Allah Knows Best.

Is it permissible to divide one mudd of fidyah between two people?

No, it is not permissible to divide one mudd between two people, because it would not count as a full meal for either of them.

What is the ruling on making up missed fasts from many years?