God clearly asked questions after the couple ate the forbidden fruit. Instead of repenting, Adam blames God; and Eve the serpent. After that (failure) to repent, they are cast out.
Eve was placed under Adam BECAUSE of her sin; that was her punishment (among other things).
God gave dominion over earth to Adam. Eve was fashioned out of his rib. He was older. All these factors gave right of passage and responsibility to the male.
Agreed. Out nature changed by the fall (indeed all of creation was changed by it). Adam and Eve became arrogant and proud, thinking they were "just like Gods." It is this arrogance and pride that inherently rejects God (as competition) in our daily lives, which is sin in its purest form.