If I change the original design after a customer has purchased it, will the changes show in the customer's purchased design?

No, if you make changes to the original design from an order after an order has been placed, it does not automatically show up in the design in the customer's order. If, however, the customer selects the  Revert to Original button, then the design will revert to the updated design (as opposed to the version that was initially purchased). 

Note: If you delete the original design from your Corjl account, and the customer selects Revert to Original, it will revert to the version of the design that was last saved.