What is the ruling on swearing on the Holy Quran and breaking that oath?

It is impermissible to swear on the Holy Quran in vain, and it is also impermissible to subject it to such an act since breaking the oath after swearing on the Quran entails an expiation.

Is it permissible for one to masturbate in order not to commit fornication?

This is called committing the milder of the two evils such as eating a carcass in fear of death, and this happens when committing fornication is very likely to occur.

Is it permissible for the big brother to force his sisters to wear Hijjab(Islamic wear) if their father didn`t do his part in this regard ?

The father`s attention should be drawn gently and politely, and the sisters should also be reminded with good words since Allah, The Almighty, says: “O ye who believe! Save yourselves and your families from a Fire whose fuel is Men and Stones.”{Al-Tahreem/6}.

Is it permissible for the person who wants to travel from the United Arab Emirates to Jordan during the day in Ramadan to break his fast before leaving his country; knowing that he had embarked on journey half an hour after dawn?

Praise be to Allah the Lord of the Worlds.

The person who made the intention to embark on journey after dawn must observe fast for that day, because it became an obligation on him before engaging on journey. However, if he experienced unbearable hardship while on travel, then he is permitted to break his fast, but he must make up for the missed day. And Allah The Almighty Knows Best.