Re: Part 1:The Imams from Ahlebayt whom Shia consider to be their divinely appointed
The following will help you
Nahjul Balaagha - Letter 58
The letter below clearly shows that Imam Ali (ra) deemed selection Hz’ Abu Bakr, Hz. Umar and Hz. Uthman (May Allah be pleased with them all) per approval and pleasure of Allah!
Verily, those who took the oath of allegiance to Abu Bakr, Umar and Uthman have sworn allegiance to me. Now those who were present at the election have no right to go back against their oaths of allegiance and those who were not present on the occasion have no right to oppose me. And so far as Shura (limited franchise or selection) was concerned it was supposed to be limited to Muhajirs and Ansars and it was also supposed that whomsoever they selected, became caliph as per approval and pleasure of Allah. If somebody goes against such decision, then he should be persuaded to adopt the course followed by others, and if he refuses to fall in line with others, then war is the only course left open to be adopted against him and as he has refused to follow the course followed by the Muslims, Allah will let him wander in the wilderness of his ignorance and schism. Nahjul Balaagha - Letter 6
Also note that when in his last days when the Prophet (saw) could not lead the prayers he appointed Hz. Abu Bakr (ra) in his place. This is very significant as it clearly implies who the Prophet (saw) want to lead the Ummah after him.
I agree with you but generally it is Imambargahs and not masjids.