Q 1: There is a Hadith that states:
It is not lawful for a person leading people in Salah (Prayer) to supplicate to himself alone without including them. If he does so, he has betrayed them.
First, what is the degree of authenticity of this Hadith? (Part No. 3; Page No. 237) Second, if the Hadith is Sahih (authentic), does it not contradict the rest of the Hadiths of the Prophet (peace be upon him) about Salah? For example, the Du`a's (supplications) to be said by the Imam (the one who leads congregational Prayer) in the singular form, such as Du`a'-ul-Istiftah (opening supplication when starting the Prayer) and after Tashahhud (testification recited in the sitting position in the second/last unit of Prayer).
The mentioned Hadith was related by
Imam Ahmad in his Musnad (vol. 5, p. 280),
Abu Dawud in his Sunan, Kitab Al-Salah (Book of Salah), chapter 43, and
Al-Tirmidhy in his Jami`, Kitab Al-Salah, chapter 148.
Al-Tirmidhy said this Hadith is Hasan (a Hadith whose chain of narration contains a narrator with weak exactitude, but is free from eccentricity or blemish).This Hadith does not contradict what you mentioned. The prohibition is specific to the Du`a' of the Imam in Witr (Prayer with an odd number of units) or Qunut (supplication recited while standing after bowing in the last unit of Prayer) at calamity or any collective Du`a' where people say "Amen". It does not mean Du`a'-ul-Istiftah, Ruku` (bowing) or such parts of Salah.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.