Firstly: 

It is prescribed to offer the night prayers in Ramadaan in congregation or individually, but it is better to do it in congregation than to do it individually. The Prophet (peace and blessings of Allaah be upon him) led his companions in offering these prayers in congregation on several nights. 

It is proven in al-Saheehayn that the Prophet (peace and blessings of Allaah be upon him) led his companions in prayer (Taraweeh) for several nights, then on the third or fourth night he did not come out to them. When morning came he said: “Nothing prevented me from coming out to you except the fact that I feared that it would be made obligatory for you.” 

Narrated by al-Bukhaari, 1129. According to the version narrated by Muslim (761): “But I feared that night prayers would be made obligatory for you and you would not be able to do them.” 

Offering Taraweeh prayer in congregation is something that is established by the Sunnah of the Prophet (peace and blessings of Allaah be upon him). The Prophet (peace and blessings of Allaah be upon him) stated the reason why he did not persist in offering this prayer in congregation, which is that he feared that it might be made obligatory. This reason cased to be applicable after the Prophet (peace and blessings of Allaah be upon him) died, because when he (peace and blessings of Allaah be upon him) died, the wahy (revelation) ceased and there was no longer any worry that it might be made obligatory. Once the reason, which was the fear of it being made obligatory, disappeared with the cessation of the wahy, then the fact that it is Sunnah to offer this prayer in congregation resumed.  

See al-Sharh al-Mumti’ by Shaykh Ibn ‘Uthaymeen, 4/78. 

Imam Ibn ‘Abd al-Barr (may Allaah have mercy on him) said: 

This indicates that praying qiyaam in Ramadaan is one of the Sunnahs of the Prophet (peace and blessings of Allaah be upon him) and is recommended and encouraged. It was not introduced by ‘Umar ibn al-Khattaab, rather he revived something that the Messenger of Allaah (peace and blessings of Allaah be upon him) loved and approved of. Nothing stopped him from doing it regularly except the fear that it might be made obligatory upon his ummah. He (peace and blessings of Allaah be upon him) was kind and compassionate towards his ummah. ‘Umar knew from the Messenger of Allaah (peace and blessings of Allaah be upon him) that the obligatory duties would not be increased or decreased after his death (peace and blessings of Allaah be upon him), so he revived this practice and enjoined it upon the people. That happened in 14 AH, and ‘Umar has the honour of being the one who revived this Sunnah.  

Al-Tamheed, 8/108, 109 

After the death of the Prophet (peace and blessings of Allaah be upon him), the Sahaabah (may Allaah be pleased with them) prayed Taraweeh in small groups and individually, until ‘Umar (may Allaah be pleased with him) united them behind a single imam. 

It was narrated that ‘Abd al-Rahmaan ibn ‘Abd al-Qaari’ said: I went out with ‘Umar ibn al-Khattaab (may Allaah be pleased with him) one night in Ramadaan to the mosque, and the people were scattered, with one man praying by himself and another with a group of men following his prayer. ‘Umar said: “I think that if I gather them behind one reader, it will be better.” Then he decided to unite them behind Ubayy ibn Ka’b. Then I went out with him on another night, and the people were praying behind their reader. ‘Umar said: “What a good innovation this is. But the prayer that they forget about and sleep is better than the one they are offering.” Narrated by al-Bukhaari, 1906. 

Shaykh al-Islam Ibn Taymiyah said, when refuting the view of those who quoted ‘Umar’s words “What a good innovation this is” as meaning that innovation (bid’ah) is permissible: 

With regard to qiyaam in Ramadaan, the Messenger of Allaah (peace and blessings of Allaah be upon him) introduced this to his ummah, and he led them in prayer for a number of nights, because at his time they used to pray in congregation and individually. But he did not persist in leading them in one congregation, lest that be made obligatory for them. When the Prophet (peace and blessings of Allaah be upon him) died, sharee’ah was established (and would not change after that). When ‘Umar (may Allaah be pleased with him) became caliph, he united them behind one imam, Ubayy ibn Ka’b, who united the people in one congregation on the orders of ‘Umar ibn al-Khattaab (may Allaah be pleased with him). ‘Umar (may Allaah be pleased with him) was one of the Rightly-Guided Caliphs, of whom the Prophet (peace and blessings of Allaah be upon him) said: “I urge you to adhere to my Sunnah and the way of the Rightly-Guided Caliphs after me; cling tightly to it.” So what he did was Sunnah but he said, “What a good innovation this is,” because it was an innovation in the linguistic sense, as they were doing something that they had not done during the life of the Messenger of Allaah (peace and blessings of Allaah be upon him), i.e., gathering to do this, but it is Sunnah in the shar’i sense.”  

Majmoo’ al-Fataawa, 22/234, 235 

For more information, please see question no. 21740 and 45781 

Secondly: 

Completing the Qur’aan in Ramadaan, whether during prayer or outside prayer, is something that is praiseworthy. Jibreel (peace be upon him) used to review the Qur’aan with the Prophet (peace and blessings of Allaah be upon him) every Ramadaan, and in the Prophet’s last year he reviewed it with him twice. 

We have discussed this in the answer to question no. 66504. 

And Allaah knows best.