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 it permissible for a husband to stop his wife from visiting her family?

A husband shouldn`t stop his wife from visiting her family, and it is disliked for him to do so because such an act makes them harbor feelings of hatred against him.

Is it permissible for a Muslim to escort his deceased disbelieving relative to his final resting place?

It is permissible provided that he doesn`t walk behind non-Islamic symbols, rather he should walk in front of 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.