1. If auto rivalries is turned on, at the start of each month rivalries will be automatically generated. The stories chosen are random and it will, I think, even generate stories you have not unlocked.
2. Matches do not trigger storylines. If you simulate a rivalry match that includes a cutscene you will not see the cutscene but the story still progresses.
To the op, it is largely a matter of preference. I switched to having RAW as the only major with Smackdown as a minor. Previously I was using both as majors but I started neglecting my Smackdown show to focus on Raw.
My advice, whether you sim or play to unlock stories have as many major shows as possible. That allows more stories to occur which makes unlocking all stories quicker.
Then if you reset universe the number of shows is up to you. Each choice has its advantages and disadvantages.