If you used a defender on a pokemon like Chansey, then used double edged and
do damge to yourself( I understand it reduces damage done to itself) but do
you still keep around to reduce your opponents damage by 20? I think the card
says dicard this card after your opponents next turn. So to make it short,
does defender reduce damage to yourself and then reduce your enemies damage?

This is another one for the "Read the Card" file.  Defender states that is reduces the first 20 damage done from attacks during a turn, and stays until the end of your opponent's turn.  So, yes, you can reduce 20 on your turn, then 20 on their turn.

Scott