After a month and a half from the end of postpartum bleeding, my wife occasionally finds a small amount of blood, and she is breastfeeding. What is this blood, and should she fast and pray when she experiences it?
This is postpartum bleeding, so she should not fast or pray until she sees the signs of purity, unless it exceeds sixty days, as the maximum duration for postpartum bleeding is sixty days, and the majority of women experience forty days. If it exceeds sixty days, it is considered irregular bleeding (Istihada). It would be advisable to consult a doctor after the sixty days. And Allah Knows Best.
My mother is 70 years old and wishes to perform Hajj and Umrah, as she has never done so before. However, she also takes care of my ill father and is afraid of falling ill herself. What should she do?
If this is the obligatory Hajj – meaning she has not previously performed Hajj or Umrah – then it is permissible for her to go even without her husband's approval. However, she must travel with a Mahram (a male guardian) or a trustworthy group of women, and she should leave someone to care for her husband and attend to his needs. And Allah Knows Best.
When is Istikharah (guidance prayer) Du`a offered, before Tasleem (saying asalamu alikum warahmatu allah wa barakatuh to end the prayer), or after it?
It is offered after Tasleem.
Does Husband fasting while being impure for having sexual intercourse with his wife at night permissible?
Yes it is permissible for you to fast because impurity isn't among valid fasting conditions, but you are obliged to perform ritual impurity in order to catch up Fajr prayer at its due time. And Allah Knows Best.