Is it permissible for a menopausal woman who has blood discharges to fast?
If she reached the age of menopause, and her menses stopped, but later she saw blood discharges that lasted a complete day and night (24) hours, then it is menstrual blood. Still, if these blood discharges lasted less than a complete day and night (24) hours, then it is not menstrual blood, but it is bleeding outside the monthly period (Istihadah), and it does not prevent her from prayer and fast, but she has to make ablution for every prayer.
Is it permissible to offer Tahajjud prayer after Witr prayer?
It is permissible to offer voluntary prayer after Witr (An odd number prayer performed between Isha and Fajr) prayer. However, it is desirable that the night prayer be ended with Witr. Therefore, it is desirable for one, who thought that he was most probably going to wake up at night in order to offer Tahajjud, to offer Witr after it, but one, who fears that he might not wake up, should offer Witr before going to bed.
Should one who doesn`t perform prayer out of laziness make it up later, and how should he/she do so?
All perfect praise be to Allah,The Lord of The Worlds He/she is obliged to make up missed prayers by offering with each obligatory prayer another one, and if he/she offers two, then it is better. And Allah Knows Best.
Is a person who is in a state of Janabah (major ritual impurity due to having a marital intercourse, ejaculation, menstruation, and post-delivery impurities) sinful if he/she goes about his/her daily life activities in that state i.e. without making Ghusl (ritual bath)?
It goes without saying that a Muslim should always be in a state of ritual purity so as to be able to perform prayers and recite Quran. It is from Sunnah (Prophetic tradition) that a Muslim hastens to make Ghusl from Janabah, but he/she is not sinful in case he/she delayed that provided that he/she doesn`t miss prayers. However, it is permissible for him/her to go about their daily activities while in a state of Janabah, but had better bathe in order not to miss any prayer.