I.
Allah, may He be Exalted, is the Creator, and everything other than Him is created, and it is not permissible to say that Allah created Himself or brought Himself into existence. Rather saying this is a grave evil because Allah, may He be Exalted, is the Creator and is not created, and He is the First before Whom there was nothing, may He be Glorified, Exalted and Sanctified.
This phrase (“He created Himself”) implies two contradictory concepts, and reconciling between them is impossible. So if you say “He created,” this means that He exists and is a creator, and if you say “Himself,” this means that He was non-existent then was created and brought into existence. Thus He is a creator and created, existent and non-existent, and this is impossible.
There is no need to quote the texts which prove that Allah is the Lord, the Creator, the One Who brought things into existence. But we will point out here an impossible question that is similar to the question that you mentioned, which is one of the questions of Iblis.
The Prophet (blessings and peace of Allah be upon him) told us that one of the specious arguments and whispers of Iblis is that he asks a person: Who created Allah?
Al-Bukhari (3276) and Muslim (134) narrated from Abu Hurayrah that the Messenger of Allah (blessings and peace of Allah be upon him) said: “The Shaytan may come to one of you and say: Who created such and such? Who created such and such? Until he says to him: Who created your Lord? If it goes that far, let him seek refuge with Allah and stop (such thoughts).”
And al-Bukhari (7296) and Muslim (136) narrated that Anas ibn Malik said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “The people will keep asking questions until they say: Allah created all things, but who created Allah?”
Ibn Battal (may Allah have mercy on him) said, narrating from someone else: If the Shaytan whispers and says: What is the problem if we say that the Creator created Himself?
The response is: This whisper contradicts itself, because saying “[He] creates” means that He already existed, and saying “Himself” implies that He did not exist. And reconciling between His both existing and not existing is a clear contradiction, because one of the conditions of the doer is that he should exist before his deed exists. Hence it is impossible that His existence could be the result of His own actions, because it is impossible to suggest that one could create another which is its very own self. Thus this specious argument is clearly refuted.
End quote from Sharh Sahih al-Bukhari, 10/343.
II.
There is nothing wrong with saying that Allah, may He be Exalted, is azali (existed from eternity) and abadi (will exist for eternity), But these are not among His divine names; rather two of His names are al-Awwal (the First) and al-Akhir (the Last), as He, may He be Exalted, says: { He is the First and the Last, the Manifest and the Hidden, and He has knowledge of all things } [Al-Hadid 57:3].
The First (al-Awwal) is the One before Whom there was nothing, as the Prophet (blessings and peace of Allah be upon him) explained it in the report narrated by Muslim (2713) from Abu Hurayrah, from the Prophet (blessings and peace of Allah be upon him), that when he wanted to sleep, he would lie down on his right side and say: “O Allah, Lord of the heavens and the earth, and Lord of the Mighty Throne, our Lord and Lord of all things, Splitter of the seed and the date stone, Revealer of the Tawrat (Torah), the Injil (Gospel) and the Furqan (Qur’an), I seek refuge in You from the evil of all things that You seize by the forelock [i.e., have full control over them]. O Allah, You are the First and there is nothing before You; You are the Last and there is nothing after You. You are the Manifest (al-Zaahir) and there is nothing above You; You are the Hidden (al-Baatin) and there is nothing beyond You. Settle our debt and spare us from poverty.”
Al-azali (the One Who exists from eternity) is the One Who has no beginning, so there was no time before Him when He did not exist, meaning that He is not created.
It says in Taj al-`Arus (27/442): The word azali is derived from the word al-azal, which refers to that which is not preceded by non-existence. What exists is of three categories, after which there is no fourth: azali abadi (that which exists from eternity to eternity), which is Allah, may He be Glorified and Exalted; la azali wa la abadi (that which neither existed from eternity nor will exist for eternity), which is this world; and abadi ghayra azali (that which will exist for eternity but did not exist from eternity), which is the hereafter – and the opposite of that (that which existed from eternity but will not exist for eternity) is impossible, because once it is proven that He existed from eternity, it is impossible to say that He will not exist for eternity.
Some scholars clearly stated that the word azali (that which existed from eternity) is not an Arabic word, or is that its root is the word yazali, which is derived from what they say of that which is ancient: “lam yazal”, from which they derived the word yazali, then the yaa’ was replaced with an alif because it is easier to pronounce, so they said azali. This is similar to the spear of Dhu Yazan [a figure in ancient Arabic literature], which they called azani [instead of Yazani], and they called a spear from Yathrib nasl athrabi (instead of yathribi). As-Saghani narrated it thus from some of the scholars. End quote.
Al-Jarjani said in At-Ta`rifat, p. 17: The word azal refers to continuation of existence throughout infinite time in the past, as the word abad refers to continuation of existence throughout infinite time in the future. Al-azali is that which is not preceded by non-existence. End quote.
See also the answer to question no. 198069 regarding the permissibility of speaking of Allah, may He be Exalted, in a manner that is not mentioned in the religious texts, if the meaning is sound.
Hence you will find that the scholars often speak about Allah describing Him as azali and qadim (existing from eternity).
At-Tahawi (may Allah have mercy on him) said in his famous book Al-`Aqidah: Allah existed with His attributes from eternity, before He created anything, and nothing was added to His attributes by bringing His creation into existence that did not exist before that. As He existed with His attributes from eternity, He will continue to exist with His attributes for eternity. End quote.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: Thus He is the First (al-Awwal) before Whom that was nothing, and He is the Last (al-Akhir) after Whom there will be nothing, so He exists from eternity and will exists for eternity without end.
End quote from Dar’ at-Ta`arud, 1/122.
What you mentioned about time being created is true, and that does not contradict what we say, that Allah exists from eternity and will exist to eternity, because you already know by now that what is meant is that His existence was not preceded by non-existence, and He, may He be Glorified, is the Creator of time and place, and He encompasses what was before and what will be after.
Ibn al-Qayyim (may Allah have mercy on him) said:
The fact that Allah, may He be Glorified and Exalted, is the First (al-Awwal) precedes the beginning of everything else, and the fact that He is the Last (al-Akhir) will remain after the end of everything else.
So His being the First precedes everything.
And His being the Last will remain after everything.
His being the Manifest (az-Zahir) means that He is exalted above all things. What is meant by being manifest is being above; the apparent part of a thing is what is on top of it, and it is surrounds what is hidden of it.
His being the Hidden (al-Batin) means that He encompasses all things, to the extent that He is closer to a thing than it is to itself.
This closeness is different from the closeness of the lover to his beloved; that is one thing, and this is something else.
These four divine names tell us that Allah encompasses everything, in two ways: in terms of time and in terms of place. So His being First and Last encompasses everything that comes before and after, and His being Manifest and Hidden encompasses everything that is apparent and hidden. There is nothing apparent but Allah is above it, and there is nothing hidden but Allah is aware of it. There is nothing first but Allah is before it, and there is nothing last but Allah is after it. The name al-Awwal (the First) means that He existed from eternity, and the name al-Akhir (the Last) means that He will exist for eternity; az-Zahir means that He is exalted and great, and al-Batin means that He is close and near.
Thus He preceded everything because He is the First, and He will remain after everything because He is the Last; He is above everything because He is the Manifest and He is close to everything because He is the Hidden. No heaven conceals another heaven from Him, no earth conceals another earth from Him, nothing that is apparent can conceal from Him anything that is hidden, but what is hidden to Him is apparent, what is unseen to Him is seen, what is far to Him is close, and what is secret to Him is open.
These four divine names refer to the pillars of Tawhid, so He is the First yet He is the Last; He is the Last yet He is the First; He is the Manifest yet He is the Hidden; and He is the Hidden yet He is the Manifest. He has always been and will always be First and Last, Apparent and Hidden.
End quote from Tariq al-Hijratayn, p. 31.
And Allah knows best.