Is it permissible to make up missed prayers at the times in which praying is disliked?
All perfect praise be to Allah,The Lord of The Worlds It is permissible to make up missed prayers at any time even that in which praying is disliked. Absolute supererogatory prayer is forbidden to be performed at the times in which praying is disliked, and this also applies to Ihram and Istikhara prayers. However, there is no harm in offering prayers in Mekah during those times. And Allah Knows Best.
Is it permissible for a woman to visit the doctor while in her `Iddah (waiting period)?
It is permissible for her to do that during day time, but it is impermissible for her to leave her house at night except for a sound reason. And Allah Knows Best.
Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings/meetings involve acts of sin?
Observing kinship ties is obligatory, and paying your blood-relatives a visit causing their sinful acts to be hindered , then you should do so. However, if their sinful acts continue while you are at their gathering and they didn't respond positively, then observing kinship ties via telephone and the like will suffice in this case. And Allah Knows
During a flight from Amman to the UAE, while we were flying over Saudi Arabia, the captain announced that it was time to break our fast, coinciding with the exact time of sunset. After I had broken my fast with my first date, the captain apologized for the earlier announcement and clarified that the correct time would be in 20 minutes. Am I required to continue fasting on that day?
Whosoever break his fast at teh very first announcement to make up the missed day of Ramadan, since Allah, The Most Exalted, Said (What means): "then complete your fast Till the night appears" [Al-Baqara/187]. And Allah Knows Best.