Praise be to Allah.
According to the correct view, you can only catch up with the prayer in jamaa’ah if you catch up with and do a rak’ah with the imaam. Merely catching up with the tashahhud or what comes before the tashahhud, but not with the rak’ah, does not count as having caught up with the prayer in jamaa’ah. I offer you my condolences for your great loss if you have caught up with nothing of the prayer except the tashahhud. The evidence (daleel) for this is the hadeeth of Abu Hurayrah (may Allaah be pleased with him), who said that the Prophet (peace and blessings of Allaah be upon him) said: “Whoever catches up with a rak’ah of the prayer has caught up with the prayer.” (Narrated by al-Bukhaari in al-Mawaaqeet, Baab man adraka min al-salaah rak’ah, 580; Muslim, Kitaab al-masaajid, 607).
This is the correct scholarly view; and it is the view of Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him), and is the view favoured by Shaykh Ibn ‘Uthaymeen (may Allaah preserve him). It is the view which is supported and strengthened by the evidence.
With regard to reciting al-Faatihah in the prayer, this is one of the most important of the arkaan al-salaah (pillars or essential parts of the prayer), and the prayer is not valid without it, because the Prophet (peace and blessings of Allaah be upon him) said: “There is no salaah for the one who does not recite the Opening of the Book.” But if a person is praying with the imaam and catches up with him during the rukoo’, for example, and he does rukoo’ without reciting al-Faatihah, this is OK, and the imaam’s recitation will count as his recitation, because of the famous hadeeth of Abu Bakrah (may Allaah be pleased with him), “May Allaah increase your keenness, but do not do this again.”
But if the imaam is clearly rushing, as you say, then the essential element of tranquillity and measured pace is not there (such as the prayer of some of the Hanafis), so the prayer offered behind him is not valid, because he is not qualified to lead it and because one of the essential elements of the prayer – that of a measured pace – is lacking. This condition is based on the famous hadeeth about the one who did not pray properly. But if it is only the case of the imam reciting relatively fast and he is able to maintain the required measured pace, in this case it is usually possible to recite Soorat al-Faatihah behind him, and if you can recite it faster, there is nothing wrong with that. But if you are not reciting al-Faatihah for another reason, such as laziness or forgetfulness, or because you joined the prayer late, then this does not affect your prayer, which is still valid, and the imaam’s recitation counts as your recitation.
The Muslim must hasten to prayer as soon as the call is given. This is one of the attributes of the believers whose hearts are attached to the mosques. Being late and only coming when the iqaamah has been given is a sign of weak faith and little desire to do good and be obedient. The Prophet (peace and blessings of Allaah be upon him) would be attending to his family’s needs, then when Bilaal gave the call to prayer, he would stop what he was doing and rush to pray. He (peace and blessings of Allaah be upon him) used to say, “Let us relax with it [the prayer], O Bilaal” and, “My delight has been made in prayer.” Whenever something distressed or worried him, he would hasten to pray. But those who are lazy and do not come to the mosque until the iqaamah has been given or during the prayer or during the tashahhud should check and rebuke themselves, for the believer’s deeds will say to him in his grave: “By Allaah, I only saw you quick to obey Allaah and slow to disobey Him.” But in the case of the munaafiq (hypocrite) and faasiq (evildoer), his deeds will say: “By Allaah, I only saw you slow to obey Allaah and quick to disobey Him.” (This is a hadeeth narrated by Imaam Ahmad. It was classed as saheeh by al-Albaani in Ahkaam al-Janaa’iz, where he also mentioned its isnaads).
Allaah described the munaafiqeen (hypocrites) as follows (interpretation of the meaning): “And when they stand up for al-salaah (the prayer), they stand with laziness” [al-Nisaa’ 4:142]. So let people pay attention to the seriousness of this matter, and let Muslim women too beware of delaying the prayer until the last moments and beware of letting worldly concerns distract them from paying attention to the provision of the Hereafter… Whoever gets used to praying regularly on time with the jamaa’ah in the mosque when the call (adhaan) is given will find that this becomes his main concern and will no longer neglect it, in sha Allaah. Whoever tries this will find deep joy, tranquillity and peace of heart and mind in the prayer. This will enable him to attain the proper sense of humility (khushoo’) when he stands before Allaah. One of the favours of Allaah to people is when a year goes by and a person never misses a single fard prayer with the jamaa’ah, except for a valid excuse. The sincere believer gets very upset if one day he sleeps and misses the prayer, or is late for the prayer because of some worldly matter, and all of the matters of this world are insignificant and fleeting. This is one of the ways in which Allaah helped His slaves to obey Him and remain steadfast.
If the iqaamah has been given, the worshipper must join the jamaa’ah straight away, even in the Fajr prayer and even if the imaam is reciting a lengthy passage, because of the hadeeth of Abu Hurayrah (may Allaah be pleased with him) who said that the Prophet (peace and blessings of Allaah be upon him) said: “Once the iqaamah for the prayer has been given, there is no prayer except the prescribed (obligatory) prayer).” (Narrated by Muslim in Kitaab Salaat al-Musaafireen, Baab 710). In this case any other prayer is not valid. For example, if a person starts a naafil prayer then the iqaamah is given, he has to stop his prayer, then make it up, if he wants to, after the (fard) prayer. If he completes the (naafil) prayer, it will not count and will be invalid.
Some scholars said that if he is nearing the end of his naafil prayer, he can complete it if he knows that he will be able to catch up with the first Takbeer (Takbeerat al-Ihraam), because the point is to hasten to join the imaam in the prayer.
And Allaah knows best.