What are all the ways a country will leave the alliance?
From limited observation, I believe that a country will leave the alliance if it is at max panic (5/5) at the time of the monthly report to the council.
Are there any other reasons why a country might leave outside of the monthly report, or if their panic level is not maxed?
Ignoring a terror report with a country will cause it to leave the council immediately, it happened to me.
I had a terror report in China, which had only 1/5 panic level at the time, and I wanted to prolong the wait as much as possible to allow a soldier of mine to heal and participate in the mission. The mission expired before the soldier was healed and China withdrew immediately.
(I then reloaded and did the mission without the soldier.)
Unlike the original, it never happened to me that a terror attack took place at the same time with another mission, so ignoring a terror event is something that you should never do, and you will never have a reason to do so.