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).

Are minors(persons under the age of puberty) obliged to fast?

A male under the age of puberty isn`t required to fast, but his guardian should order him to fast if he was able to endure the hardship until he gets used to it.

What is the ruling on having brotherly ties between a strange man and a strange woman, and is the former considered a Mahram(unmarriageable) to the latter?

Such a relation between a strange man and a strange woman is forbidden in Islam because it involves forbidding what Allah has allowed by marriage, and allowing what Allah has forbidden such as looking and the like. Such a relation doesn`t render any act lawful between them, thus they are forbidden to look at each other, have a Khulwah(seclusion), and travel together.

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.