Is it permissible for the doctor to uncover the hair of a female patient out of necessity?
It is only permissible for him to do so when there is no female doctor, and he should examine the patient in the presence of one of her Mahrams (unmarriageable kin).
I joined a savings club and was scheduled to receive the payout first. My friend asked to take my place in line, so I requested 50 dinars in exchange for letting her have my turn. What is the ruling on this?
This is impermissible, as it involves taking money without providing something of value in return according to Islamic law. And Allah The Almighty Knows Best.
I underwent a procedure for a molar pregnancy (hydatidiform mole), and after that, the bleeding continued for more than two months, during which I did not pray. Do I need to make up the missed prayers?
If you have previously experienced postpartum bleeding, the duration of the current bleeding is treated the same as your previous postpartum period. Any bleeding that exceeds this duration must be considered irregular bleeding (istihada), and you must make up the missed prayers. If this is the first time, then any bleeding beyond sixty days is considered irregular bleeding. You must perform ghusl, pray, and make up the prayers for the days beyond the sixty-day limit. And Allah Knows Best.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.