A first look into the presented verses
Let's try to take a look at the three verses presented:
The first verse (2:97) is indicating that Jibreel or Gabriel has revealed the Qur'an or transmitted it to Muhammad (). This should be an important remark here!
The second verse (2:98) might create confusion, because Michael or Mikhaeel and Gabriel (Jibreel) have been quoted beside the angels and messengers, as if they were different from both! Note that in Arabic language this might be an indication for the important "position" or "rank" they may have while they either are messengers or angles to create a distinction between them and the majority! So basically from a linguistic perspective we could say they either are indeed different or part of humans or angels!
The third verse (66:4) indicates that Gabriel (Jibreel) is a protector of the Prophet() beside the righteous believers!
Tafseer of the verses
From tafsir ibn Kathir abou the two first verses (2:97-98):
Imam Abu Ja'far bin Jarir At-Tabari said, "The scholars of Tafsir agree that this Ayah (2: 97-98) was revealed in response to the Jews who claimed that Jibril (Gabriel) is an enemy of the Jews and that Mika'il (Michael) is their friend.'' Al-Bukhari said, "Allah said,
(Whoever is an enemy of Jibril (let him die in his fury)).
'Ikrimah said, "Jibr, Mik and Israf all mean, worshipper, while il means, Allah''. Anas bin Malik said, "When 'Abdullah bin Salam heard of the arrival of the Prophet in Al-Madinah, he was working on his land. He came to the Prophet and said, "I am going to ask you about three things which nobody knows except a Prophet. What will be the first portent of the Hour What will be the first meal taken by the people of Paradise Why does a child resemble its father, and why does it resemble its maternal uncle" Allah's Messenger said, (Jibril has just told me the answers.) 'Abdullah said, "He (i.e. Jibril), among all the angels, is the enemy of the Jews." Allah's Messenger recited the Ayah,
(Whoever is an enemy to Jibril (Gabriel) (let him die in his fury), for indeed he has brought it (this Qur'an) down to your heart). Allah's Messenger then said, (The first portent of the Hour will be a fire that will bring together the people from the east to the west; the first meal of the people of Paradise will be the caudate lobe of the liver of fish. As for the child resembling his parents: If a man has sexual intercourse with his wife and his discharge is first, the child will resemble the father. If the woman has a discharge first, the child will resemble her side of the family.) On that 'Abdullah bin Salam said, "I testify that there is no deity worthy of worship except Allah and you are the Messenger of Allah." 'Abdullah bin Salam further said, "O Allah's Messenger! The Jews are liars, and if they should come to know about my conversion to Islam before you ask them (about me), they will tell a lie about me." The Jews came to Allah's Messenger , and 'Abdullah went inside the house. Allah's Messenger asked (the Jews), ("What kind of man is 'Abdullah bin Salam") They replied, `He is the best among us, the son of the best among us, our master and the son of our master.' Allah's Messenger said, (What do you think if he would embrace Islam) The Jews said, "May Allah save him from it." Then 'Abdullah bin Salam came out in front of them saying, "I testify that none has the right to be worshipped but Allah and that Muhammad is the Messenger of Allah." Thereupon they said, "He is the evilest among us, and the son of the evilest among us." And they continued talking badly about him. Ibn Salam said, "This is what I feared, O Messenger of Allah!.''' Only Al-Bukhari recorded this Hadith with this chain of narration. Al-Bukhari and Muslim recorded this Hadith from Anas using another chain of narration.
(source qtafsir where you might find more on the meaning of "il")
Also read in sahih al-Bukhari: in the book of tafsir the hadith of Anas quoted above.
In the context of verse (66:4) ibn Kathir (see here in qtafsir) apparently seemed to have quoted this hadith from sahih Muslim.
I certainly could do more research to find fatwas or tafisir books addressing the issue of what Gabriel/Jibreel actually is! But I'd rather prefer a logical explanation.
Logical explanation based on the Qur'an
In the Qur'an we learn:
Allah chooses from the angels messengers and from the people. Indeed, Allah is Hearing and Seeing. (22:75)
Which means that only humans or angels can be messengers! So here already we can exclude that Gabriel might be a Jinn. Note that in the context of Jinn, we know that there are believers, but from the Qur'an we know that they believed by the intermediate of human messengers (in the qur'an at least Musa/Moses() and Muhammad() have been quoted).
Now verse (2:97) informs us that Gabriel has revealed the Qur'an to Muhammad(), so already here we could say if Gabriel was human, what is the use of an other human prophet or messenger who is only an intermediate? Especially as we know that Muhammad was not been taught or instructed by a human:
And We certainly know that they say, "It is only a human being who teaches the Prophet." The tongue of the one they refer to is foreign, and this Qur'an is [in] a clear Arabic language. (16:103)
Let's take an other verse:
Say, "If there were upon the earth angels walking securely, We would have sent down to them from the heaven an angel [as a] messenger." (17:95)
Here again we see that angels are excluded to be messengers for human beings!
From all of this we can conclude that Gabriel indeed is neither a Jinn nor human, which means that he is an angel!
Note on the hadith you quoted before editing:
As at the end of the hadith you may read:
"That was Gabriel who came to teach the people their religion."
Abu Hurraira only knew about the identity of the visitor or questioner when Muhammad () revealed it to the present audience or witnesses among the sahaba . This can also be concluded from other ahadith quoting appearances of Jibreel in a human shape such as the hadith of 'Omar ibn al-Khattab.