Is it permissible for the children of a deceased father to settle his debt from the Zakah (obligatory charity) money due on them?

It is impermissible to use the Zakah of one`s money for settling the debts of the deceased. However, children should settle the debts of their deceased parents out of filial piety. 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.

Does the clipping of nails nullify ablution?

No, it does not, and it is preferable to wash the hands afterwards.

How should a person who is afflicted with continued major ritual impurity (incontinence of urine, bleeding outside the monthly period) perform ablution?

Such a person should make ablution for every prayer after its due time and after removing impurities from their body and outfit, and should place a fresh diaper whenever needed so as for the impurity not to spread out. They should also perform prayer immediately even if incontinence of urine continued, and should repeat the aforementioned for every obligatory prayer, then perform optional prayer as much as they want.