The Qur'an was revealed in progression, as you mentioned, with verses from the same chapter being revealed not necessarily in sequence. The Qur'an was revised in its entirety with Gabriel once a year, and twice in the last year prior to the death of the Prophet ﷺ. This typically took place in the month of Ramadan.
In the following hadith narrated by Abu-Huraira (and similar others, e.g., this hadith by 'Aisha), the Qur'an annual revisions are mentioned:
حَدَّثَنَا خَالِدُ بْنُ يَزِيدَ، حَدَّثَنَا أَبُو بَكْرٍ، عَنْ أَبِي حَصِينٍ، عَنْ أَبِي صَالِحٍ، عَنْ أَبِي هُرَيْرَةَ، قَالَ كَانَ يَعْرِضُ عَلَى النَّبِيِّ صلى الله عليه وسلم الْقُرْآنَ كُلَّ عَامٍ مَرَّةً، فَعَرَضَ عَلَيْهِ مَرَّتَيْنِ فِي الْعَامِ الَّذِي قُبِضَ، وَكَانَ يَعْتَكِفُ كُلَّ عَامٍ عَشْرًا فَاعْتَكَفَ عِشْرِينَ فِي الْعَامِ الَّذِي قُبِضَ فِيهِ
Narrated Abu-Huraira: Gabriel used to repeat the recitation of the Qur'an with the Prophet (ﷺ) once a year, but he repeated it twice with him in the year he died. The Prophet (ﷺ) used to stay in I'tikaf for ten days every year (in the month of Ramadan), but in the year of his death, he stayed in I`tikaf for twenty days.
— Sahih al-Bukhari 4998
The Prophet ﷺ did the same with his companions, some times even with the companions specifically named by Gabriel as in the case of Ubayy ibn Ka'b:
حَدَّثَنَا هَدَّابُ بْنُ خَالِدٍ، حَدَّثَنَا هَمَّامٌ، حَدَّثَنَا قَتَادَةُ، عَنْ أَنَسِ بْنِ مَالِكٍ، أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم قَالَ لأُبَىٍّ " إِنَّ اللَّهَ أَمَرَنِي أَنْ أَقْرَأَ عَلَيْكَ " . قَالَ آللَّهُ سَمَّانِي لَكَ قَالَ " اللَّهُ سَمَّاكَ لِي " . قَالَ فَجَعَلَ أُبَىٌّ يَبْكِي .
Anas reported Allah's Messenger (ﷺ) as saying to Ubayy b. Ka'b:
Allah has commanded me to recite the Qur'an to you. He said: Did Allah mention me to you by name? He (the Holy Prophet) said: Allah made a mention of your name to me. (On hearing this) Ubayy b. Ka'b wept.
— Sahih Muslim 799 a
In the history of the compilation and documentation of the Qur'an:
قال أبو عبد الرحمن السلمي: قرأ زيد بن ثابت على رسول الله صلى الله عليه وسلم في العام الذي توفاه الله فيه مرتين، وإنما سُميت هذه القراءة قراءة زيد بن ثابت، لأنه كتبها لرسول الله صلى الله عليه وسلم وقرأها عليه وشهد العرضة الأخيرة، وكان يقرئ الناس بها حتى مات، ولذلك اعتمد عليه أبو بكر وعمر في جمع القرآن، وولاه عثمان كتابة المصاحف.
— NOTE: My own translation, so treat with care:
Abu Abd al-Rahman al-Salami said that Zayd ibn Thabit recited [the Qur'an] to the Messenger of Allaah ﷺ in the year in which he died twice. This recitation is named Zayd ibn Thabit's recitation because he wrote it to the Messenger of Allaah ﷺ and recited it to him as the last revision, and he recited it to the people until he died. So Abu Bakr and 'Umar appointed him [Zayd ibn Thabit] on the compilation of the Qur'an, and 'Uthman appointed him to manage the documentation of the mus-hafs [Qur'an manuscripts].
— IslamWeb, Compilation of the Qur'an by Abu Bakr (Arabic only)
As for the "complete and perfect now" part of the question, Surat Al-Ma'idah 5:3 says exactly so: "This day I have perfected for you your religion and completed My favor upon you and have approved for you Islam as religion." This verse was revealed on the day of the Farewell Sermon given by the Prophet ﷺ on the 9th of Dhu al-Hijjah, 10 AH:
حَدَّثَنَا الْحَسَنُ بْنُ الصَّبَّاحِ، سَمِعَ جَعْفَرَ بْنَ عَوْنٍ، حَدَّثَنَا أَبُو الْعُمَيْسِ، أَخْبَرَنَا قَيْسُ بْنُ مُسْلِمٍ، عَنْ طَارِقِ بْنِ شِهَابٍ، عَنْ عُمَرَ بْنِ الْخَطَّابِ، أَنَّ رَجُلاً، مِنَ الْيَهُودِ قَالَ لَهُ يَا أَمِيرَ الْمُؤْمِنِينَ، آيَةٌ فِي كِتَابِكُمْ تَقْرَءُونَهَا لَوْ عَلَيْنَا مَعْشَرَ الْيَهُودِ نَزَلَتْ لاَتَّخَذْنَا ذَلِكَ الْيَوْمَ عِيدًا. قَالَ أَىُّ آيَةٍ قَالَ {الْيَوْمَ أَكْمَلْتُ لَكُمْ دِينَكُمْ وَأَتْمَمْتُ عَلَيْكُمْ نِعْمَتِي وَرَضِيتُ لَكُمُ الإِسْلاَمَ دِينًا}. قَالَ عُمَرُ قَدْ عَرَفْنَا ذَلِكَ الْيَوْمَ وَالْمَكَانَ الَّذِي نَزَلَتْ فِيهِ عَلَى النَّبِيِّ صلى الله عليه وسلم وَهُوَ قَائِمٌ بِعَرَفَةَ يَوْمَ جُمُعَةٍ.
Narrated 'Umar bin Al-Khattab: Once a Jew said to me, "O the chief of believers! There is a verse in your Holy Book Which is read by all of you (Muslims), and had it been revealed to us, we would have taken that day (on which it was revealed as a day of celebration." 'Umar bin Al-Khattab asked, "Which is that verse?" The Jew replied, "This day I have perfected your religion For you, completed My favor upon you, And have chosen for you Islam as your religion." (5:3) 'Umar replied,"No doubt, we know when and where this verse was revealed to the Prophet. It was Friday and the Prophet (ﷺ) was standing at 'Arafat (i.e. the Day of Hajj)"
— Sahih al-Bukhari 45
Note that the above verse was not the last verse revealed. Most scholars say Surat Al-Baqarah 2:281 was revealed after it as the last verse to be revealed in the Qur'an.