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.
What is the ruling when the duration of the monthly period varies, and sporadically increases?
If the period didn`t exceed (15) days from the first seeing of blood discharges until its end including the times of temporary cessation, then it is menses. But, if the period exceeded (15) days, then the woman experiencing this is Mostahadah (bleeding outside the time of monthly period), and should abide by her regular period, and consult one of the scholars.
Should water be put in the deceased`s mouth and nose while washing him/her?
Putting water in the mouth and nose of the deceased while washing him is impermissible.
What is the ruling on making up for missed fasting after the second half of Sha`ban (the month before Ramadhaan)?
One is obliged to make up for missed fasting before the start of next Ramadhaan, and regardless of offering it during the first, or the second half of Shab`an. This is because the prohibition mentioned in the Hadith is for offering absolute voluntary fasting in the second half of Sha`ban. And Allah Knows Best.