Assalaamu alaykum. What will be the ruling about these cases: 1) A person joins the imam in prayer when the imam has already finished the first salam (to the right side) 2) Is it an excuse to pray at home (or alone in the masjid) that the imam prays so quickly that it becomes quite impossible to complete Surah Al-Faatihah and the first tashahhud behind him?

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 is His slave and Messenger.

The prayer ends with the first tasleem (to the right) according to most of the scholars, and the second tasleem is a recommended Sunnah. Hence, if the praying person’s ablution is invalidated after the first tasleem and before the second one, then his prayer is valid. If the imaam makes the first tasleem, then following him in prayer ends. The Shaafiʻi scholar An-Nawawi wrote, "If the imaam makes the first tasleem at the end of the prayer, then the obligation of following him in the prayer ends for both the latecomer and the one who joined the congregational prayer at its very beginning because his prayer has ended (by that first tasleem)..." [Al-Majmooʻ]

Hence, based on this scholarly view, the latecomer should not join the congregation prayer after the imaam makes the first tasleem. If he does, then his prayer is invalid and he must repeat it. Shaykh Ibn ʻUthaymeen wrote:

"If the latecomer recites takbeerat al-ihraam (opening takbeer) before the imaam makes the first tasleem, then he has joined the congregation prayer properly and his prayer counts. However, the latecomer should not join the congregational prayer after the imaam already made the first tasleem. The scholars stated that the prayer of the latecomer who joins the prayer after the imaam makes the first tasleem does not count and that he must repeat it because by making the first tasleem, the imaam has begun to end the prayer. Hence, it is not valid for a latecomer to intend following the imaam in the congregational prayer while the imaam has begun to end the prayer..." [Ash-Sharh Al-Mumtiʻ]

As for your question about whether the person is excused for missing the congregational prayer at the mosque and praying at home instead if the imaam prays very quickly, the religious ruling in this regard depends on whether such quickness prevents the required tuma’neenah (tranquility in each position of the prayer) or not. Tuma’neenah is one of the pillars of the prayer without which the prayer is void. If the imaam’s quickness hinders the fulfillment of the prerequisite of tuma’neenah in prayer, then his prayer as well as that of the led is invalid. It is obligatory to replace him with another imaam if possible. If it is hard to replace him, then the led person should find another mosque where he can avail himself of the great rewards of attending the prayer in congregation. If this is difficult, then there is no harm in performing the prayer at home in this case. The praying person should try to perform the prayer in congregation with his household so as to avail himself of the great rewards of congregational prayer.

However, if the imaam’s quickness does not hinder the fulfillment of the prerequisite of tuma’neenah in prayer, it is advisable that you should not deny yourself the rewards of attending the congregational prayer at the mosque since the Imaam’s prayer is valid. Verily, attending the congregational prayer at the mosque is greatly rewardable, and a group of scholars even held that it is obligatory. Hence, the Muslim should keenly observe the congregational prayer at the mosque. please refer to fatwa 82538.

The criterion of the due attainment of tuma’neenah in prayer, as Ibn Hajar (Al-Haytami) asserted in his book Tuhfat Al-Muhtaaj, is that each body part settles in the appropriate position before one moves from one position to another.

The Hanbalis advised that the minimum of it (tuma’neenah) is that one observes tranquility even if this is for a short period. Al-Mardaawi said, "This is the correct view of the (Hanbali) School, and it was also said, 'It is to the extent of being able to say the obligatory mention of Allaah (in a particular position of the prayer).'" For more benefit, please refer to fatwa 180233.

As for missing the first tashahhud when joining the congregational prayer, you should know that the first tashahhud is not among the pillars of the prayer; rather, it is among its obligations. The fact that the praying person being led in the prayer failed to complete the recitation of the first tashahhud does not affect the validity of his prayer. The same ruling applies to the recitation of the Faatihah chapter (no. 1) when the worshipper being led in the prayer fails to complete the recitation of the Faatihah chapter after the imaam, his prayer is not invalid according to the view of most of the scholars; they held that the recitation of the imaam is sufficient for the led.

Ibn Qudaamah wrote, "In brief, it is not obligatory on the led to recite the Faatihah after the imaam in audible or inaudible prayers, as asserted by Imaam Ahmad on the authority of the group of his students. The same opinion was held by Az-Zuhri, Ath-Thawri, Ibn ʻUyaynah, Maalik, Abu Haneefah, and Is-haaq..." [Al-Mughni]

The opinion adopted by Islamweb is that the led are obliged to recite the Faatihah after the imaam in inaudible and audible prayers alike, otherwise their prayer is void. Hence, based on this view, if you cannot recite the Faatihah after this imaam given his quickness, then you may attend the congregational prayer in any other mosque, as previously advised, so as to avail yourself of the great rewards and virtue of attending the congregational prayer. If you cannot do so, then you are advised to perform the prayer in congregation at your house; you should not opt for performing the prayer individually. The rewards of praying in congregation can still be attained even when the person is not performing the prayer in the mosque.

Allaah knows best.