Regarding slot playing like klopp, i don't think that is true. Gakpo said to four four two magazine last January that under slot they were much more precise and patient and played shorter passes and that under klopp they would kick it long then fight for it.
From what I read about klopp, I hos last season, when building from the back with a low sitting defence, he would play 4-3-3. One 6 and two 8s. When building with a high sitting defence they would play 1-2-2-5-1, with one fullback pushing up as a winger and moving another into midfield. One of their more common formation changes was supposedly 1-3-1-5-1, with a holding player dropping into the defence when playing against two forwards.
When pressing, klopp supposedly went man to man. Against back 4s I read he gets into the man to man system. They did this by pushing a right winger on opposition cb, rb on opposition lb and cb in opposition left winger. When pressing with a low sitting defence, they supposedly set up in a 1-4-1-4-1, controlling the center and forcing the opposition out wide.
Perhaps slot benefited from the tough, fighting winning mentality klopp instilled in the team, but I think from what I have been reading that he did his own thing.