Adam and Eve were to multiply themselves and subdue the lands & creatures of the wilderness to incorporate them harmoniously into the template of the garden. They were to rule over creation according to their best judgement—the core meaning of being created in God's image—without need to till the ground because the garden provided food on its own.
The curses in Genesis 3:14-19 are reversals of the original plan of creation. God curses the snake (an unsubdued creature) to go on its belly (it was presumably like a cobra that goes about upright like a human); he curses Eve to multiply with difficulty (menstruation, pregnancy, birth, etc.) and to subject herself to Adam; and he curses Adam to labor hard to provide for his family.
The point of Genesis 3:20 thus seems to be that Eve forfeits her status as an equal counterpart to Adam, and her primary contribution becomes motherhood (1 Timothy 2:15)—a contribution that should not be minimized given that most believers come to faith through their parents, hence the inclusion of the command to honor father and mother with the first four 'vertical' commandments.
Note: The words *'êzer kəneḡdō* (translated as "helper" in Genesis 2:18) literally mean "counterpart"; *ishah* (translated as "woman" in Genesis 2:23) means “from man”; and *khavva* (translated as "Eve" in Genesis 3:20) means “life.”
