What is the law of Friday prayer and Eid Fitr and Qurban prayers during the absence of Imam and what is their quality?
Your question will be answered in two parts:
1. Laws of Friday Prayer
Friday Prayer is an important duty of Islamic duties, in which God has commanded to it in His Book and has said: ﴿يَا أَيُّهَا الَّذِينَ آمَنُوا إِذَا نُودِيَ لِلصَّلَاةِ مِنْ يَوْمِ الْجُمُعَةِ فَاسْعَوْا إِلَىٰ ذِكْرِ اللَّهِ وَذَرُوا الْبَيْعَ ۚ ذَٰلِكُمْ خَيْرٌ لَكُمْ إِنْ كُنْتُمْ تَعْلَمُونَ﴾; “O you who have believed! When the call of prayer is made on Friday, hasten to the remembrance of God and give up all trading, that is better for you, if you only knew” and according to the Islamic Mutawatir narrations, it has two sermons and two Rak’ahs, and its property which has likely caused it to be commanded to in the Book of God is the existence of two sermons for it, and therefore the Friday Prayer will not be performed without two sermons and consequently without anyone who recites the two sermons, and these two sermons should be done before the prayer and while standing; as one of our companions informed us and said:
«قُلْتُ لِلْمَنْصُورِ: مِنْ أَیْنَ قُلْتَ إِنَّ خُطْبَةَ الْجُمُعَةِ مِنْ قِیامٍ؟ قالَ: لِقَوْلِ اللّهِ تَعالیٰ: ﴿وَإِذَا رَأَوْا تِجَارَةً أَوْ لَهْوًا انْفَضُّوا إِلَيْهَا وَتَرَكُوكَ قَائِمًا ۚ﴾، قُلْتُ: صَدَقْتَ وَ مِنْ أَیْنَ قُلْتَ إِنَّها قَبْلَ الصَّلاةِ؟ قالَ: لِقَوْلِ اللّهِ تَعالیٰ: ﴿فَإِذَا قُضِيَتِ الصَّلَاةُ فَانْتَشِرُوا فِي الْأَرْضِ﴾، قُلْتُ: صَدَقْتَ إِنَّ عِنْدَكَ عِلْمَ الْكِتابِ، قالَ: عِنْدِي عِلْمٌ مِنَ الْكِتابِ وَ إِنَّ عِلْمَ الْكِتابِ عِنْدَ الْمَهْدِيِّ وَلَوْ جِئْتُمُوهُ لَعَلَّمَكُمْ أَكْثَرَ مِمّا عَلَّمْتُكُمْ، فَسَكَتَ ساعَةً ثُمَّ قالَ: إِنَّما مَثَلِي فِیكُمْ كَمَثَلِ رَجُلٍ وَجَدَ مَعْدِناً مِنَ الذَّهَبِ فَأَخَذَ مِنْهُ قِطْعَةً كَبِیرَةً فَجاءَ بِها إِلَی التُّجّارِ فَقالَ لَهُمْ: إِنِّي وَجَدْتُ مَعْدِناً مِنَ الذَّهَبِ فَأَیُّكُمْ یُعِینُنِي عَلَی اسْتِخْراجِهِ لِیَكُونَ لَهُ فِیهِ نَصِیبٌ؟ فَیَقُولُونَ لَهُ: كَیْفَ نَعْلَمُ أَنَّكَ قَدْ وَجَدْتَهُ؟! فَیُرِیهِمُ الْقِطْعَةَ الْكَبِیرَةَ فَیَعْلَمُونَ أَنَّهُ قَدْ وَجَدَهُ فَیُعِینُهُ مَنْ یَشاءُ وَ یَخْذُلُهُ مَنْ یَشاءُ»; “I said to Mansoor: How did you say that the sermon of Friday prayer should be done while standing? He said: for the word of Almighty God that <When they see a transaction or a diversion, they hasten to it and leave you standing>, I said: You are right, and how did you say it is done before prayer? He said: for the word of Almighty God that <And when the prayer has been concluded, disperse within the land>, I said: You are right, you have the knowledge of the Book, he said: I have some knowledge of the Book, and Mahdi has the knowledge of the Book, and if you come to him, he will teach you more than what I taught you, then he was silent for an hour and then he said: Indeed my example among you is that of a man who found a gold mine, so he took a large piece of it and brought it to the merchants and told them: Indeed I have found a gold mine, so which one of you will help me mining that to have a share in it? So they will say to him: How do we know that you have found it? So he shows them the big piece, so they know he has found it, so whoever wants to help him would do so and whoever wants to leave him would do so”.
Just as these two sermons must include remembering of God and saluting His Prophet and reading His Book and inviting to righteousness and promoting good and prohibiting evil to be better than diversion and trading for Muslims; as God has said: ﴿وَإِذَا رَأَوْا تِجَارَةً أَوْ لَهْوًا انْفَضُّوا إِلَيْهَا وَتَرَكُوكَ قَائِمًا ۚ قُلْ مَا عِنْدَ اللَّهِ خَيْرٌ مِنَ اللَّهْوِ وَمِنَ التِّجَارَةِ ۚ وَاللَّهُ خَيْرُ الرَّازِقِينَ﴾; “And when they see a transaction or a diversion, they hasten to it and leave you standing, say: What is with God is better than diversion and transaction, and God is the best of providers”, and therefore whenever the two sermons include false speaking, similar to praising Taughut and inviting to evil and promoting bad and prohibiting good, do not suffice, rather staying and listening them are Haram; as God has said: ﴿وَالَّذِينَ لَا يَشْهَدُونَ الزُّورَ وَإِذَا مَرُّوا بِاللَّغْوِ مَرُّوا كِرَامًا﴾; “And those who do not testify to falsehood, and when they pass near ill speech, they pass by with dignity” and has said: ﴿وَالَّذِينَ هُمْ عَنِ اللَّغْوِ مُعْرِضُونَ﴾; “And they who turn away from ill speech” and has said: ﴿وَإِذَا سَمِعُوا اللَّغْوَ أَعْرَضُوا عَنْهُ﴾; “And when they hear ill speech, they turn away from it” and has said: ﴿فَاجْتَنِبُوا الرِّجْسَ مِنَ الْأَوْثَانِ وَاجْتَنِبُوا قَوْلَ الزُّورِ﴾; “So avoid the uncleanliness of idols and avoid false statement” and has said: ﴿وَقَدْ نَزَّلَ عَلَيْكُمْ فِي الْكِتَابِ أَنْ إِذَا سَمِعْتُمْ آيَاتِ اللَّهِ يُكْفَرُ بِهَا وَيُسْتَهْزَأُ بِهَا فَلَا تَقْعُدُوا مَعَهُمْ حَتَّىٰ يَخُوضُوا فِي حَدِيثٍ غَيْرِهِ ۚ إِنَّكُمْ إِذًا مِثْلُهُمْ ۗ إِنَّ اللَّهَ جَامِعُ الْمُنَافِقِينَ وَالْكَافِرِينَ فِي جَهَنَّمَ جَمِيعًا﴾; “And He has already sent down to you in the Book that when you hear the verses of God are denied and ridiculed, so do not sit with them until they enter into another conversation, you would then be like them, indeed God will gather the hypocrites and disbelievers in Hell all together”; Especially given that following oppressors and sinners in the prayer is not permissible except for Taqiya  and with this description, it is forbidden to attend a Friday prayer held by the Imamate of a tyrannical ruler or an imam appointed by him or a sinner imam; as one of our companions informed us and said:
«سَأَلْتُ الْمَنْصُورَ عَنْ صَلاةِ الْجُمُعُةِ، فَقالَ: فَرِیضَةٌ، قُلْتُ: لا یُقِیمُهَا الْیَوْمَ إِلّا كُلُّ جَبّارٍ عَنِیدٍ، فَقالَ: لا تُصَلِّ مَعَهُمْ فَإِنَّ مَنْ صَلّیٰ مَعَهُمْ فَهُوَ مِنْهُمْ إِلّا أَنْ یَتَّقِيَ مِنْهُمْ تُقاةً وَ كانَ عِنْدَهُ رَجُلٌ فَقالَ الرَّجُلُ: أَما كانَ السَّلَفُ الصّالِحُ یُصَلُّونَها مَعَهُمْ مُنْذُ قُبِضَ رَسُولُ اللّهِ صَلَّی اللّهُ عَلَیْهِ وَ آلِهِ وَ سَلَّمَ؟! قالَ: بَلیٰ وَلٰكِنَّهُمْ كانُوا یُصَلُّونَ بَعْدَها أَرْبَعَ رَكَعاتٍ»; “I asked Mansoor about Friday prayer, so he said: It is a duty, I said: today it is not held unless by a disobedient tyrant, then he said: do not pray with them, for whoever prays with them, he is from them, unless he is doing Taqiya, and there was a man beside him, so the man said: did the righteous predecessors not perform it with them since the messenger of God peace be upon him and his household passed away?! He said: yes, but after that they performed four Rak’ahs (Dhuhr prayer)”.
As another of our companion informed us and said:
«دَخَلْتُ عَلَی الْمَنْصُورِ فِي یَوْمِ جُمُعَةٍ، فَقالَ لِي: صَلّیٰ هٰؤُلاءِ جُمُعَتَهُمْ؟ قُلْتُ: نَعَمْ، فَقالَ: لا وَاللّهِ ما صَلُّوا وَلٰكِنْ لَغَوْا وَ نَقَرُوا، قُلْتُ: إِنَّ فُلاناً مِنْ أَصْحابِكَ یَدْخُلُ فِیهِمْ فَیُصَلِّيها مَعَهُمْ، قالَ: وَ یَفْعَلُ؟! قُلْتُ: نَعَمْ، قالَ: أَلا یَخافُ أَنْ یَخْسِفَ اللّهُ بِهِمُ الْأَرْضَ أَوْ یُنْزِلَ عَلَیْهِمْ رِجْزاً مِنَ السَّماءِ وَ هُوَ فِیهِمْ؟!»; “I reached in Mansoor’s presence on a Friday, so said to me: did they do their Friday prayer? I said: yes, then he said: no, I swear by God they did not, but they made a noise and pecked, I said: a person from your helpers goes among them and performs it with them, he said: he does such an action?! I said: yes, he said: Is he not afraid of God to pull them into the earth or send them a torment from the sky, while he is among them?”.
Yes, it is obligatory to attend a Friday prayer held by the imamate of the Caliph of God on earth or by an imam appointed by him or a righteous imam from the preparers of his advent, and it will be performed at the time of Dhuhr prayer and with a community of believers; because based on Mutawatir narrations, the Friday prayer is a substitute for Dhuhr prayer and therefore, it is not permissible to do it before or after the time of Dhuhr prayer; as “Friday prayer” is the prayer of a community of the believers, and therefore if the number of attendants is so low that they do not count as a community of believers, then the “Friday Prayer” will not be performed and a community of believers, according to a narration from Ahl al-Bayt, is a number of five people and this is reasonable; because one of them is an imam and the other is a caller who calls for prayer and is usually a non-imam and the other three are considered to be the minimum of a community, and therefore the condition of holding Friday prayer can be considered the presence of at least five believers; as one of our companions informed us and said:
«سَأَلْتُ الْمَنْصُورَ عَنْ أَقَلِّ عَدَدٍ یَنْعَقِدُ بِهِمُ الْجُمُعَةُ، فَقالَ: الْإِمامُ وَ الْمُنادِي لِلصَّلاةِ وَ ثَلاثَةُ نَفَرٍ مِنَ الْمُؤْمِنینَ»; “I asked Mansoor about the fewest number of people that Friday Prayer is held by them, so he said: the imam and caller for prayer and three people from the believers”.
As in every city there will not performed more than one Friday prayer; because the appearance of the Book of God and Islamic narrations is that there was only one Friday prayer in the city of the prophet peace be upon him and his household, unless it was a large city, and attending the Friday Prayer after hearing its call gets a group of people into hardship and difficulty; because in this case they can hold another Friday prayer; given that God has said: ﴿يُرِيدُ اللَّهُ بِكُمُ الْيُسْرَ وَلَا يُرِيدُ بِكُمُ الْعُسْرَ﴾; “God intends for you ease and does not intend for you hardship” and has said: ﴿مَا يُرِيدُ اللَّهُ لِيَجْعَلَ عَلَيْكُمْ مِنْ حَرَجٍ﴾; “God does not intend to make difficulty for you”. With this description, if the necessary conditions for performing another Friday prayer were not met, it will not be obligatory upon them anymore; just as if, it is not obligatory for other excused peoples such as patients, elderlies and passengers.
2. Laws of Eid prayer
There is disagreement among Muslims about obligation of Eid prayer; respecting that the Shiite, the Hanafiyyah and some of the Shafi’i consider it obligatory objectively, and the Hanbali considers it obligatory sufficiently and the Malikiyah and most of the Shafi’i consider it as emphasized recommendation, but the truth is that just as the Friday prayer, it is obligatory objectively, and similar to that, it will be removed from those who have an excuse; because God, -contrary to most people’s assumption- has mentioned it in His book and the prophet has persevered it in his Sunnah and he has not abandoned it in any year; as one of our companions informed us and said:
«سَأَلْتُ الْمَنْصُورَ عَنْ صَلاةِ الْفِطْرِ وَ صَلاةِ الْأَضْحیٰ، فَقالَ: فَرِیضَتانِ، قُلْتُ: إنَّهُمْ یَقُولُونَ لَوْ كانَتا فَرِیضَتَیْنِ لَنَزَلَ فِیهِمَا الْقُرْآنُ، قالَ: وَ هَلْ یَعْلَمُونَ الْقُرْآنَ كُلَّهُ؟! قُلْتُ: لا، قالَ: فَلَوْ رَجَعُوا إِلَی الَّذِینَ یَعْلَمُونَ الْقُرْآنَ كُلَّهُ لَعَلَّمُوهُمْ، ثُمَّ سَكَتَ، فَقُلْتُ: جُعِلْتُ فِداكَ وَ هَلْ نَزَلَ فِیهِمَا الْقُرْآنُ؟ قالَ: نَعَمْ إِنَّ اللّهَ تَعالیٰ یَقُولُ: ﴿وَلِتُكْمِلُوا الْعِدَّةَ وَلِتُكَبِّرُوا اللَّهَ عَلَىٰ مَا هَدَاكُمْ﴾ وَ ذٰلِكَ صَلاةُ الْفِطْرِ وَ یَقُولُ: ﴿فَصَلِّ لِرَبِّكَ وَانْحَرْ﴾ وَ ذٰلِكَ صَلاةُ الْأَضْحیٰ»; “I asked Mansoor about Fitr and Qurban prayers, then he said: they are obligatory, I said: they say that if they were obligatory they would be revealed in the Quran, he said: do they know the whole of Quran? I said: no, he said: then if they have referred to those who know the whole of Quran, they would have taught them, then he became silent, then I said: may I be your sacrifice, have Quran been revealed in about them? He said: yes, Almighty God says: <And to complete the number (of the days of Ramadan), and to glorify God that He has guided you> and that is the Fitr prayer, and He says: <So pray to your Lord and sacrifice> and that is the Qurban prayer”!
Of course, the conditions of this prayer’s obligation are similar to the conditions of Friday prayer’s obligation, and its differences with Friday prayer is in five things:
According to the Islamic Mutawatir narrations, the time of Eid prayer is from sunrise to noon, and therefore it will not be replaced by noon prayer.
According to the Islamic Mutawatir narrations, it is Sunnah to perform Eid prayer outside the city and under the sky, except in Mecca where it is better to perform it at Masjid al-Haram.
Third; the sermon
According to the Islamic Mutawatir narrations, the sermon is recited after the Eid prayer and reciting it before the Eid prayer is an invented-heresy by the Bani Umayyad.
Fourth; Adhan and Iqamah
According to the Islamic Mutawatir narrations, before the Eid prayer, Adhan and Iqamah are not recited and reciting Adhan and Iqamah before that is an invented-heresy by the Bani Umayyad.
According to the Islamic Mutawatir narrations, there are more Takbirat in the Eid prayer and this is a law that the Book of God refers to; as one of our companions informed us and said:
«سُئِلَ الْمَنْصُورُ عَنْ صَلاةِ الْعِیدِ، فَقالَ: ﴿وَلِتُكْمِلُوا الْعِدَّةَ﴾ یَعْنِي شَهْرَ رَمَضانَ ﴿وَلِتُكَبِّرُوا اللَّهَ عَلَى مَا هَدَاكُمْ﴾ یَعْنِي اثْنَتَيْ عَشْرَةَ تَكْبِیرَةً ﴿وَلَعَلَّكُمْ تَشْكُرُونَ﴾ یَعْنِي الْقُنُوتَ وَ لَیْسَ بِواجِبٍ»; “Mansoor was asked about the Eid prayer, so he said: <And to complete the number> means the month of Ramadan <And to glorify God that He has guided you> means twelve Takbirs <And perhaps you be thankful> means Qunut and is not obligatory”.
Therefore, more Takbirats in the Eid prayer is obligatory based on ﴿وَلِتُكَبِّرُوا اللَّهَ عَلَى مَا هَدَاكُمْ﴾ and it is understood from Islamic narrations and saying of Mansoor Hashemi Khorasani may God protect him that the Sunnah is Twelve Takbirs; in this way that five Takbirs are in the first Rak’ah and four Takbirs are in the second Rak’ah in addition to Takbiratul Ihram and two Takbirs of Ruku, and it is recommended that among the Takbirs, a Qunut be recited in praising God and appreciating Him for the blessing of guidance and the blessings of the cattle.