What is the ruling on having brotherly ties between a strange man and a strange woman, and is the former considered a Mahram (unmarriageable) to the latter?
Such a relation between a strange man and a strange woman is forbidden in Islam because it involves forbidding what Allah has allowed by marriage, and allowing what Allah has forbidden such as looking and the like. Such a relation doesn`t render any act lawful between them, thus they are forbidden to look at each other, have a Khulwah (seclusion), and travel together.
A woman didn't fufill fasting of the month of Ramadan two years ago, due to pregnancy and breastfeeding, at the time being she is making up the missed Ramadan. What is the ruling of Sharia? And what is due on her?
Whosoever break the fast during Ramadan or didn't fast at all due to health concerns, is obliged to make up the missed fasts whenever she could so long as making up missed Ramadan didn't extend to the coming one, and if next one arrived without fulling fasting the missed one, the ransom is 60 grams for each missed day (Equals 60 piasters to one Dinar for each day). And Allah The Almighty Knows Best.
Is it permissible for someone to make a supplication to Allah to marry an already married, or an engaged woman?
No, it isn`t, and the person who does so is sinful before Allah, The Almighty.
What is the ruling when the monthly period when exceeding (15) days?
The maximum duration of menstruation is 15 days. If it exceeds this, it is considered irregular bleeding (Istihada). In this case, one must perform ghusl (Ritual bath) after menstruation, pray, and make wudu for each prayer after its time begins. She should return to her previous menstrual habit regarding duration and timing. For example, if her period used to be from the 20th to the 27th of each month, this is considered her menstruation period. Anything before or after this is Istihada, during which she only leaves prayer and fasting for the habitual menstrual days, then performs ghusl and resumes praying.