When you publish the card, if it is your own image or design then you own the rights. You could go down the route of getting a copyright however it probably won't be cost-effective. Even if you own the copyright are you willing to pay a lawyer to pursue a claim against someone? It is likely if you have a popular design, it will be copied. This could be on any number of sites and by people around the world. It's illegal but unfortunately, it happens a great deal. I have heard that an ongoing joke in China says, "a copyright means the right to copy".
The answer to your second question is yes sometimes this does happen. However, if you are trying to market to retailers, contact Zazzle and see if their terms and conditions allow you to do this.