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.
Is ablution invalidated when blood comes out of the nose, or a wound?
Blood coming out of the nose, or a wound does not invalidate ablution, but it is preferable to make ablution as a way out of the scholars disagreement in this regard.
Is it permissible for a person, who vowed to offer two Rak`ahs (units of prayer) everyday for Allah`s sake, to do so after the Fajr (dawn) prayer?
In principle, a vow should be fulfilled as it was originally intended, and if its time lapses then, it should be fulfilled at any time because of the vow. Therefore, it is permissible to offer them after the Fajr prayer, and there is no prohibition in doing so. However, it is preferable for him to offer these two Rak`ahs at some other time to avoid the disagreement amongst the scholars in this regard.
Is it permissible to give my zakat to my grandson who is studying at university, while I am living abroad and my money is in my home country? Also, is it permissible to assign my father the responsibility of distributing the zakat to the people in my home country?
It is permissible to give zakat to a son whom his father is supporting if he is an adult and in good health, because his father is not obligated to support him, making him one of the poor among the Muslims. However, the student who may receive zakat is one who is diligent in seeking beneficial knowledge for the Muslims and is religious. Zakat should be paid in the country where the money is located, and appointing your father to distribute the zakat in the country where the money is is the correct approach. And Allah Knows Best.