When you buy designs they are kept in your download center. Log in and you will see the download center at the top center of the page. all designs that you bought are kept there and you can download again. Free designs are not in download center only bought designs. Hope this helps you.
The design numbers listed were offered for Free from Cute members. That section of the website is now closed. Those designs downloaded for free will not be in your Download Center. If you have other designs you purchased individually, you might be able to access those by logging in under your old account.
If you sign in to your earlier account (using the old avatar and password) and click on the arrow next to the 'Hi, your name' in the top bar, you get a pull down menu like the one in the picture. Click on Download Center or on Your Orders and you can find all orders - not the freebies and not the designs you downloaded during a membership - and can download them again.