What type of illness that permits breaking the fast in Ramadan?

It is an illness that, if one fasts, there is a fear it may lead to their death or cause unbearable hardship.

Is it permissible to combine and shorten prayers at home before traveling?

It is impermissible to combine and shorten prayers at home before traveling because a person is called a traveler once he/she departs the borders of his/her country, and offering prayer that way was originally intended to exempt a traveler. And Allah Knows Best.

Is it permissible to offer prayer at home, or should it be offered in the mosque?

All perfect praise be to Allah, The Lord of The Worlds.                                                                                                                                                                      It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. 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?

Ruling on bringing a cooked sheep as an Aqiqah

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.