What is to recite after al-Fatiha in prayer?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

1) It is Sunnah for one who prays to recite some Surah or a portion of al-Qur'an after al-Fatiha in the two Raka'hs of al-Fajr, al-Jumu'ah and the first two Raka'hs of al-Zuhr, al-Asr, al-Maghrib and al-Isha and the optional prayers.
It is recorded in al-Bukhari and Muslim from Abu Qatadah (Radiya Allahu Anhu) who reported that "The Prophet (Sallallahu Alaihi wa Sallam) used to recite al-Fatiha and some Surah in each of the 1st two Rak'ahs, of al-Zuhr prayer and only al-Fatiha in the third and fourth Raka'hs, and recited a lengthy portion in the first one. The Prophet (Sallallahu Alaihi wa Sallam) used to do the same in al-Asr and al-Fajr prayers" .
So, it is sufficient for a worshipper to recite al-Fatiha only in all Raka'hs of prayer and the prayer is valid.
A sound Hadith recorded by al-Bukhari from Abu Hurairah (Radiya Allahu Anhu) reads: "He (the Prophet (Sallallahu Alaihi wa Sallam)) used to recite in every Rak'ah. So, we made you hear what he let us hear, and we concealed what he kept secret. Then, if you do not recite an additional portion to al-Fatiha, that is sufficient; and if you do, that will be better" .
As for Sujud al-Sahu in this case, you are not required to do such Sujud if you intentionally leave the additional portion. And, according to some scholars, it is needed to do Sujud al-Sahu if you forget the additional portion of al-Qur'an after al-Fatiha.
2) It is sufficient to recite any portion of al-Qur'an after al-Fatiha even it is one verse.
It is proved, according to al-Daraqutni with a strong chain of narrators, that Ibn Abbas (Radiya Allahu Anhu) recited al-Fatiha and one verse from al-Baqara Surah for each Rak'ah.
3) Your prayer is valid with the Imam. As for what you mentioned concerning reciting the last verses of some Surahs is not proved, we state that Ibn al-Qayyim said: 'That is not reported from him (Sallallahu Alaihi wa Sallam)' . He added: 'Maybe he recited the same Surah in two Raka'hs' . In fact, Ibn al-Qayyim 's statement is related to what is better and favorite not to validity of a prayer.
As for differentiation between Nafil (optional prayers) and Fard (obligatory prayers), we say that authentic narrations from the Prophet (Sallallahu Alaihi wa Sallam) state that he ordered an Imam to avoid long recitation while performing Fard as recorded by al-Bukhari and Muslim from Ibn Masoud (Radiya Allahu Anhu) who reported, "A man said to Allah's Messenger (Sallallahu Alaihi wa Sallam): "I intentionally go late to prayer due to long recitation of the Imam in al-Fajr prayer. The Prophet (Sallallahu Alaihi wa Sallam) became too angry more than any other previous time and said: "O people! Some of you make others dislike prayer. So, if one of you leads the people in prayer, he should be easy, for among the people are the weak, sick, aged, and one who has to go for some of his needs" .
In brief, it is permissible to recite any Qur'anic portion after al-Fatiha, whether he is performing Fard or Nafil. Then, the Prophet (Sallallahu Alaihi wa Sallam) used to recite the whole Surah in one Rak'ah, sometimes in two Raka'hs or repeat the same Surah in the second Rak'ah. In simple words, one has to take such a question easily, since there is no hard-and-fast rule in this concern.
Allah knows best.