Al-Bukhaari (4500) and Muslim (6890) narrated from Abu Moosa that the Prophet, sallallaahu ʻalayhi wa sallam, said:
“Two gardens of silver, their vessels and everything in them; and two gardens of gold, their vessels and everything in them. And there will be nothing between the people and their seeing their Lord, may He be blessed and exalted, except the veil of pride concealing His Face in the Paradise of ‘Adan.”
Why does Allaah, The Exalted, not remove His veil of pride? Does He remove his veil of light? How many veils does he have? What more can I know about Allaah/His characteristics, for example, He has a Cloak, Fingers...

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

Your question, “Why does Allaah, The Exalted, not remove his veil of pride?” is wrong to begin with; verily, Allaah, The Exalted, is not to be questioned as to what He does or does not do. There is none like Allaah, and no comparisons should be drawn for Him. The governing principle in the sound belief in this regard is stated in the verse that reads (what means): {He is not questioned about what He does, but they will be questioned.} [Quran 21:23]

As for your question, “Does Allaah remove His veil of light?”; it has been narrated on the authority of Suhayb that the Prophet, sallallaahu ʻalayhi wa sallam, said, “When those deserving of Paradise will enter Paradise, Allaah, The Exalted, will ask, ‘Do you wish Me to give you anything more?’ They will say, ‘Have You not brightened our faces? Have You not made us enter Paradise and saved us from Hellfire?’” He (the Messenger of Allaah) said, “He (Allaah) will lift the veil, and of all the things given to them, nothing will be dearer to them than the sight of their Lord, The Mighty and Glorious.” [Muslim]

As for the question about the number of veils that Allaah has taken between Him and His slaves, Shaykh ʻAli ibn Khudhayr Al-Khudayr wrote:

“The veils that Allaah, The Exalted, took between Him and His creation: This issue was addressed by some Sunni scholars as one of the issues related to the Face of Allaah and is classified under this heading, as Ad-Daarimi did in his refutation of the views of the tyrant Bishr Al-Mareesi (from the Jahmiyyah sect). It has been established in the Quran and the authentic Sunnah that Allaah has veils. As for the supportive evidence from the Quran, Allaah, The Exalted, says (what means): {And it is not for any human being that Allaah should speak to him except by revelation or from behind a veil...} [Quran 42:51]

As for the evidence from the Sunnah, it has been narrated on the authority of Suhayb that the Prophet, sallallaahu ʻalayhi wa sallam, said, “When those deserving of Paradise will enter Paradise ... He (Allaah) will lift the veil...” [Muslim] The hadeeth stated that the dwellers of Paradise will see Him then, which proves that the issue of the veil of Allaah comes under the heading of seeing His Face.

Moreover,  Abu Moosa narrated a marfoo’ hadeeth (i.e. a hadeeth directly attributed to the Prophet, sallallaahu ʻalayhi wa sallam) that reads, ‘His veil is light.’ Ad-Daarimi and others cited a mawqoof report (i.e. stopped at the level of the Companion) on the authority of Ibn ʻUmar that reads, ‘Allaah, The Exalted, veiled Himself from His creation with four veils: fire, light, darkness, and light.’

The hadeeth reported on the authority of Ibn ʻUmar was supported by another mawqoof hadeeth on the authority of ‘Abdullaah ibn ‘Amr that says that Allaah has veiled Himself with four veils. Moreover, it has been narrated on the authority of Zuraarah ibn Abi Awfa in a marfoo' hadeeth that the Prophet, sallallaahu ʻalayhi wa sallam, asked Jibreel (Archangel Gabriel), ‘Have you seen your Lord?’ Jibreel replied, ‘Between me and Him there are seventy veils of light.’ [Ad-Daarimi] This hadeeth was cited by Al-Bayhaqi in his book Al-Asmaa’ wa As-Sifaat, and by Ibn Khuzaymah in At-Tawheed.

In brief, it is established that Allaah is veiled from His creation and that He has many veils between Himself and His slaves; the exact number and nature of those veils is unknown (to us). There are some reports saying that these veils are four, and others suggested that they are seventy; Allaah knows best. As for the nature of such veils, some reports stated that Allaah has veils of fire, light and darkness, and He knows best.” [Az-Zinaad fi Sharh Lumʻat Al-Iʻtiqaad]

With regards to the Cloak of Allaah that was mentioned in some ahaadeeth, one of which is, “Pride is My ridaa' (cloak) and greatness is My izaar (lower garment)...”, the Permanent Committee for Islamic Research and Fataawa in KSA was asked:

“The marfoo' hadeeth reported by Abu Hurayrah in saheeh Muslim reads, 'Pride is My ridaa' and Majesty is My izaar, and I shall throw him who shares with Me regarding one of them into Hellfire.' How can we understand this hadeeth? Is it permissible to ascribe the attribute of having a cloak or lower garment to Allaah or do we in this case have to interpret these attributes?” The answer reads:

“Al-Khattaabi, may Allaah have mercy upon Him, in his explanation of Sunan Abu Daawood (book), said that the meaning of the hadeeth is that Pride and Majesty are two Attributes of Allaah, The Exalted, that are confined to Him Alone; no created being can share them with Him, and no created being should show them; because a created being is described with modesty and humility. With regard to the saying that Allaah has a cloak and a lower garment, it is, as Al-Khattaabi said - though Allaah knows best - a simile that means that just as no one shares another human's cloak or lower garment with him, no one can share with Allaah the Attributes of Pride and Majesty.”

Please refer to fataawa 88429, 90390 and 92488 about some of the attributes of Allaah.

Allaah knows best.