It looks very strange, Coupons data are always stored in "cart_rule" Table.
Can you please confirm whether the "remaining coupon (cart rule)" are displayed at the Admin cart rule page or not.
Also verify that you are checking the correct DB, as you told that you have many Coupons created at your store and there are no entries in the cart_rule table.
Regarding : What's an easy way of changing these dates in mysql?
=>If you want to make any changes in the table, then the easies way is to use the queries