Is it permissible to sleep after the time of the Fajr prayer has begun, with the intention of waking up shortly afterward?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
If a person sleeps after the time of Fajr has begun, with the second adhān, then sleeping is forbidden (ḥarām) at that point, unless he is certain that he will wake up and perform the prayer within its time. And Allah Almighty knows best.
Is it permissible to trim the eyebrows if they become excessively long?
Trimming eyebrows is permissible if they become excessively long, yet; it is reprehensible. And Allah Knows Best.
What is the ruling of islamic law on statutory (legal) interest awarded by a court?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible for a person in whose favor a judge has awarded statutory (legal) interest to deduct the actual litigation costs incurred in filing and prosecuting the lawsuit from these interest funds.
Any remaining balance beyond those actual expenses must be returned to the party upon whom the payment was imposed, provided that party is a specific individual. If, however, the paying party is a public entity or state institution, the remaining amount should be spent on the general public interests of Muslims—such as schools, mosques, roads, and similar public utilities shared by the community—or given in charity (Sadaqah) to the poor and needy. And Allah Almighty knows best.
Is the marriage, which lacks a valid legal contract, a guardian, and a court registration, valid?
It is incumbent that a valid marriage contract be concluded in the presence of a guardian and two trustful witnesses, and it should be registered in the court to protect the rights of the wife. Actually, a valid marriage contract is what differentiates between sound marriage and fornication.