A lot of what you're carrying on about, or stumbling over, is just semantics. "league average player" in the WAR context means league average starter. "league average player" does not mean the mean talent of everybody who laces them up, for whatever amount of time, in the big leagues. In reality in baseball there is functional overlap between all levels, and a lengthy regress of talent. If we want, we can expand our circle from MLB starters to all MLB contributors, to AAA, AA, A, rookie ball and Carlos Gonzalez will go from a 50% percentile talent all the way to a 99.99th percentile talent. Your low PA cutoff was just making the circle too big. And your high PA cutoff was making it too small (teams use platoons, players get hurt - all "qualified" players taken together do not give us enough innings to play a full MLB season, not even close!)
As for your last question, 162 games or 650 PA I believe, although I'm not 100%. See Jonathan Schoop last year for a close to ideal real world illustration. Go to the "Value" tab of his Fangraphs page.
It's good that you're questioning things and thinking about things critically.