Yeah, the manager is a factor. Usually a rather small one, given that most managers are just "OK". And no manager is good enough or influential enough to overcome negatives in ALL of the other factors.
Klopp is a better manager than Rodgers, yeah. We all know that. But if managers were THAT important and influential, then one could start to make a case for Claudio Ranieri being a better manager than Klopp. He's top of the league! With Leicester.
Yeah, well that goes without saying. Due to the investment in football the playing field isnt fair at all besides between some of the top teams. So winning something makes it extremely difficult bar a cup run etc.
But when you look at the clubs battling relegation, a managers quality could see you avoid relegation instead of going down.
That has happened several times.
On the last point. No, you cant.