What is required of a traveler or a sick person if they broke their fast and then their excuse ceased?

If a traveler settles or a sick person recovers after having broken their fast, it is recommended for them to refrain from eating and drinking for the rest of the day, but it is not obligatory.

What is the ruling if someone wipes his head then shaves it - must he repeat it?

If a person performing ablution shaves his head after wiping it during ablution, he is not required to re-wipe his head or repeat the ablution. And Allah the Almighty knows best.

Does the time of Fajr (dawn) prayer start after the first Athaan (call for prayer), or the second one?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                                    The time of Fajr prayer starts after the second Adhan. And Allah Knows Best.

Is it acceptable to perform the Aqiqah for a male child by slaughtering and distributing the first sheep, and bringing the second one cooked from the restaurant?

 

Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.

It is permissible to slaughter the first sheep with the intention of Aqiqah (the newborn's sacrificial offering) and distribute it entirely [uncooked], and to slaughter the second sheep and have it cooked at a restaurant to bring home for the household. However, it must be noted that it is obligatory to give some portion of the Aqiqah in charity to the poor, even if it is a small amount, though it is preferable to send the food cooked to them.

Buying a pre-cooked, ready-made sheep from a restaurant does not suffice as an Aqiqah. However, if an agreement is made with the restaurant to explicitly slaughter a sheep with the intention of Aqiqah for the newborn, and then cook it afterward, this is permissible.

In conclusion, slaughtering the sheep and distributing it with the intention of Aqiqah is permissible, and through it, the foundational prophetic tradition (Sunnah) is fulfilled. As for simply buying a cooked sheep from a restaurant that was not specifically slaughtered with the intention of Aqiqah, it will not count as such. Conversely, if the restaurant owner is commissioned (Wakala) to handle both the slaughtering and the cooking as an Aqiqah, it is valid—provided that a portion of it, even if small, is given in charity, which is estimated to be approximately half a kilogram of meat. And Allah the Almighty knows best.