To host you need to have meet everyone and research Printing Press first. I assume the runner up is an other civ who met either one of the requirements before everyone else.
No; while the original host is decided that way, as the world council changes eras there is a new vote for host.
In all cases though it's both the host and whoever has the most votes (host excluded) at the time that proposals can be made and in the case of tie, it appears to be player order that breaks it.
For the first council, the following has been my experience when an AI was first to reach it:
A. Forbidden Palace built by an AI (but not the above host) : Both the AI that's hosting and the AI with FP will be proposing resolutions until the council changes eras (unless city with FP taken over / host wiped out)
B. FP not built : In this case, every non-host is tied for 1 vote each (until either the world council advances to "welcomes city states" or someone builds the FP) ; resolved in favor of the human being able to propose due to the tie being broken by player order. Which the human will be prompted to do before the host due to this being done in player order.
When I was first to reach this and hosted:
A. Forbidden Palace built by an AI: Both myself and the AI with Forbidden Palace will be proposing resolutions until the council changes eras (unless city with FP taken over)
B. Forbidden Palace not built: In this case all the AIs are tied for 1 vote each (until either the world council advances to "welcomes city states" or someone builds the FP) ; that tie broken in favor of whichever AI's turn immediately follows the human. The human still prompted to propose before the AI.