What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
What is the ruling on insulin injections?
Insulin injections do not break the fast because they are administered under the skin and do not reach the body cavity (jauf).
What is the ruling on combining (jam‘) and shortening (qaṣr) prayers when travel is repeated on a daily basis?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
A traveler is permitted to avail himself of the concessions of shortening and combining prayers once the distance of his journey reaches (81 km) or more. This is on condition that his journey is a permissible one, and that he begins shortening and combining only after setting out and passing beyond the built-up limits of the city in which he resides, or beyond the administrative boundaries of the area where he lives.
Accordingly, if the distance between one's workplace and one's place of residence reaches the distance upon which the concession of combining and shortening depends, then combining and shortening are permissible once he leaves the built-up area, even if this travel is repeated on a daily basis.
Ash-Shirbīnī (raḥimahullāh) said: "It is permissible to combine Ẓuhr and ‘Aṣr, either as an early combination (taqdīm) in the time of the first, or as a delayed combination (ta'khīr) in the time of the second; and likewise Maghrib and ‘Ishā' — either early, in the time of the first, or delayed, in the time of the second — during a long, permissible journey, in following [the Prophet's practice]. As for the delayed combination, it is established in the two Ṣaḥīḥs from the ḥadīth of Anas and Ibn ‘Umar (raḍiyallāhu ‘anhum). As for the early combination, Ibn Ḥibbān and al-Bayhaqī graded it ṣaḥīḥ from the ḥadīth of Mu‘ādh, and at-Tirmidhī graded it ḥasan." [Quoted with slight abridgment from Mughnī al-Muḥtāj, Vol.1/p.529]
The conditions for combining and shortening have already been explained in Fatwa No. 14 and Fatwa No. 842. And Allah, the Exalted, knows best.
Can a person required to give kaffarah feed it to their own family members?
The kaffarah must be given to the poor and needy who are not financially dependent on the one giving the kaffarah.
If a person feeds it to their own family members, it does not count as kaffarah, and their obligation remains unfulfilled.