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.
                             
                        
                            
                                
                                After I had prayed, I found a drop of menstrual blood and was unsure when that happened; what is the ruling on my prayer and fast?
                                All praise is due to Allah, peace and blessing be upon the Messenger of Allah.
Your prayer and fast are correct ,and you should not make up for them, and we ask Allah, The Almighty, to duely reward you for these acts of worship. Your menses start once you see blood because the situation remains as such unless there is certainty or predominant conjecture, as per the Fiqh maxim: "Certainty is not removed by doubt". And Allah Knows Best.
                             
                        
                            
                                
                                Is it permissible to make up for missed fast after the beginning of the second half of the month of Sha`ban (the month before Ramadan)?
                                Yes, it is permissible, but one who had missed fasts should hasten to make up for them. As for the Hadith mentioned in this regard, the prohibition is for offering absolute voluntary fasting. And Allah 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.