Is it permissible to make ablution without answering the call of nature after waking up?
                                Washing front and back openings (Istinjaa`) is not a condition for the validity of ablution because it is done to remove Najaasah (impurity), thus if there is no Najaasah, there is no need for that, and then it is permissible to make ablution without answering the call of nature. However, prayer is invalidated if there is Najaasah on the anus ,or the front opening, so it (Najaasah) must be removed for the prayer to become valid, and not the ablution.
                             
                        
                            
                                
                                Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
                                A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
                             
                        
                            
                                
                                Is the Saum (Fasting) of someone who ate and drank forgetfully while offering fasting of oath expiation invalidated?
                                Whosoever eats, or drinks forgetfully is exempted by Allah; therefore, he/she shouldn`t break their fast whether it was obligatory, non-obligatory, or expiatory. And Allah Knows Best.
                             
                        
                            
                                
                                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.