Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
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.
If the menses lasted for over than 9 to 10 days, considered menstruation?
If the duration of the bleeding does not exceed fifteen days, it is all considered menstruation, even if the usual pattern changes.
The supplication: "O Allah, we do not ask You to avert the decree, but we ask You for kindness in it." Is this Supplication Halal or Haram?
We are obliged to supplicate Allah, so when supplicating we comply with Allah's Ordainments, and making supplications that are reported in the authentic texts is better. And Allah Knows Best.