Is it permissible to pay the Zakah to my grandmother who lives alone in a rented accommodation with bad conditions noting she had three sons. 

Zakah cannot be given to ascendants (parents and grandparents). It is obligatory for her sons and grandsons to provide for her, and they would be sinful if they fail to do so. And Allah Knows Best.

What is the ruling on making a false oath?

Making false oath indulges its maker in Hell; therefore, he is obliged to repent, ask Allah for forgiveness and make an expiation; feeding 10 poor or providing clothes for them and in case of incapability of doing any of latters he should fast for 3 consecutive days. And Allah Knows Best. 

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.

I became ill while fasting and had to take an intramuscular injection twice on two consecutive days during Ramadan. I continued my fast as usual, as I was told that intramuscular injections do not break the fast. Please advise me so that I can know the correct ruling.