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.

What is the ruling on taking part in an online contest to win a prize of 100,000 dollars in return for answering thirty questions in different fields of knowledge and science? Knowing that there is no drawing of lots; only one or two cards are purchased to commence the contest.

Is it permissible for a woman to swear a false oath by the Holy Quran to please her distrustful husband?

It is forbidden for a person to do so since this is considered a false oath (Yamaan Ghamoos) which dips (Yaghmos) its taker in Hellfire if he/she doesn`t seek repentance, and he/she has to expiate for that oath. You should avoid any doubtful acts so that your husband`s chest remains clear from the evil suggestions of the devil.

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.