i believe the last part of your question needs to be answered first. you have asked under whose authority are the jinns cast out?.
Allah the mighty and sublime says in the quran;
....and Allah has power over all things.[48:21]
And to Allah belongs the dominion of the heavens and the earth, and Allah has power over all things.[3:189]
there are numerous other verses that states that it is Allah with Whom all power is and all the authority. no one, no created being can do anything except by the will of Allah. the jinns are cast out also by the will and might of Allah (swt). to believe otherwise is a form of 'shirk'. may Allah protect you and me from such.
exorcism (ruqya) is allowed in islam but we must understand though there are numerous authentic hadith relating to performing/reciting ruqya, we must always try to follow and adhere to what the prophet (P.B.U.H) has thaught us to be the best way in doing something.
Narrated Ibn `Abbas:
Allah's Messenger (ﷺ) said, "Seventy thousand people of my followers will enter Paradise without accounts, and they are those who do not practice Ar-Ruqya and do not see an evil omen in things, and put their trust in their Lord.
Reference : Sahih al-Bukhari 6472
these people will enter paradise without being brought to account because of the perfection of their Tawheed, their complete trust in Allah and their independence from people. the one who asks for ruqyah from others is not included in that seventy thousand who will enter paradise without being brought to account, due to the shortfall in their trust in Allah, because asking for ruqyah involves a kind of humiliation and need of the raaqi (the one who performs ruqyah), and part of complete trust and Tawheed is that the muslim should not ask people for anything.
Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said:
These people are praised because they did not ask anyone to recite ruqyah for them, and ruqyah is a type of du’aa’, so they did not ask others to pray for them. The phrase “and they did not recite ruqyah” which is mentioned in the hadeeth is a mistake (on the part of the narrator), for their ruqyah for themselves and for others is a good deed. The Prophet (peace and blessings of Allaah be upon him) recited ruqyah for himself and for others; he did not ask anyone to recite ruqyah for him. His reciting ruqyah for himself and others was like his making du’aa’ for himself and others; this is something that is enjoined, for all the Prophets asked of Allaah and prayed to Him, as Allaah tells us in the stories of Adam, Ibraaheem, Moosa and others.
Majmoo’ al-Fataawa, 1/182
Shaykh Ibn Baaz (may Allah have mercy on him) said:
This hadeeth indicates that not asking is better, just as not using cautery is better, but when there is a need for it, there is nothing wrong with asking for ruqyah or using cautery, because the Prophet (blessings and peace of Allah be upon him) told ‘Aa’ishah to ask for ruqyah for an illness that had befallen her, and he told the mother of the children of Ja‘far ibn Abi Taalib (may Allah be pleased with him), whose name was Asma’ bint ‘Umays (may Allah be pleased with her), to seek ruqyah for them. This indicates that there is nothing wrong with that when there is a need to ask for ruqyah. End quote.
Majmoo‘ Fataawa Ibn Baaz, 25/118-119
it is always better to perform ruqya on yourself as it is a type of dua and as believers, dua is the strongest weapon we have against anything. in the same vein as the saying 'prevention is better than cure', islam teaches us the best practice. reciting the daily zikr, performing the obligatory duties, avoiding sin, much remebrance of Allah and having complete trust in Allah is the protection and remedy we can have against anything.
Allah knows best.