War in Heaven doesn’t happen unless you have Leviathans DLC. Even if you do, it’s somewhat unlikely to occur. Naturally in the course of a game a Fallen Empire will usually Awaken (this is based on your end-game start date, the default number of 2400 is too high if you aren’t a beginner, imo), though the year it will happen is randomized to a fair extent.
All Fallen Empires have only 1 ethic (spiritualist/materialist, xenophile/xenophobe, and 1 machine empire) and if the opposite ethic FE awakens the war in heaven starts and the galaxy divides into three factions. Some races will submit to the awakened empires and join their faction, while some will form a 3rd faction, the League of Non-Aligned Powers.
Sadly I know all this because the achievement to win the war in heaven as the leader of the non-aligned powers is one of the more irritating ones to get.
Edited to add: If a second FE does not awaken within 10 years to the day of the first awakening, no further FE’s will awaken to begin the war in heaven.