Is it permissible for a woman to apply Kuhul(Halal eyeliner) and go out wearing it?

Kuhul is part of a woman`s adornment, and it is forbidden for her to show that to anyone other than a husband , or a Mahram(Unmarriageable kin).

Is the patient who is on dialysis twice a week obliged to observe fasting? Is his fasting on the days he is on dialysis considered valid or not?

Praise be to Allah the Lord of the Worlds.

Dialysis invalidates fasting because of the nutritional fluid used in that procedure, as pointed by specialized doctors, in addition to causing bodies to enter the stomach. Therefore, it is recommended that the patient who had undergone dialysis to observe fasting for the rest of that day, if he is able to, and make up for it after Ramadan. However, if the doctor told him that fasting, at any time, would damage his health, then he must break his fast and pay a ransom against every missed day. We pray that Allah blesses this person with recovery. And Allah The Almighty Knows Best.

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.

Has Islam made it incumbent on us to give our children particular names?

No, it hasn`t, but it is desirable that we give them good names.