Why does "malfacilajn problemojn" take the accusative even though it follows "por"?
It's the object of "analizi".
It doesn't directly follow it; the thing that por "controls" is just the word analizi.
Could someone use "kielo" instead of "metodo" they seem like they mean the same thing, they are both a way of HOW to do something
A method is indeed a way of doing something, but it's a way that has been thought about previously. Not all kieloj are methodical.