Should a Muslim pray in congregation at a mosque where he does not feel at ease with his prayer, due to the imam's rushing of the prayer and the muezzin's invoking blessings upon the Prophet after the adhan?

It is not obligatory to go to the mosque if the distance exceeds that from which the call to prayer can be heard, such as twenty kilometers or twenty minutes by bus. One can pray in congregation with some brothers, or individually if that is not possible. There is no harm in going to a distant mosque if the Imam performs his prayer with ṭumaʾnīnah (tranquility/composure). A Muslim should not be concerned about the mu'adhin raising his voice with the salat upon the Prophet, peace be upon him, nor about the haste of the mosque-goers in performing the Sunnah prayer. Rather, a Muslim should pray whenever he finds a congregation of Muslims, regardless of their gender, language, or country.