What should a worshiper who has forgotten a pillar of prayer do?

If he/she remembered the forgotten pillar before offering it in the following Rakah (unit of prayer), then he/she should offer it, and continue the prayer, then offer Sujood As-Sahw (Prostration of forgetfulness) at the end of it. But, if he/she remembered that forgotten pillar after having offered it in the following Rakah, then he/she should offer a new Rakah in its (the one in which he forgot a pillar) place, and offer Sujood As-Sahw.

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 a woman shaking hands with the brothers of her husband`s father, and not putting on Islamic wear before them?

The aforesaid are strangers to the their brother`s son`s wife, therefore, it is forbidden for her to take off her Islamic wear before them, or to shake hands with them.

Is the patient who is on dialysis twice a week obliged to observe fasting? Is his fasting on the days he is on dialysis considered valid or not?

Praise be to Allah the Lord of the Worlds.

Dialysis invalidates fasting because of the nutritional fluid used in that procedure, as pointed by specialized doctors, in addition to causing bodies to enter the stomach. Therefore, it is recommended that the patient who had undergone dialysis to observe fasting for the rest of that day, if he is able to, and make up for it after Ramadan. However, if the doctor told him that fasting, at any time, would damage his health, then he must break his fast and pay a ransom against every missed day. We pray that Allah blesses this person with recovery. And Allah The Almighty Knows Best.