An avatar is the picture beside a username. A stickie is a square picture that appears at the bottom of a post.
I received an email saying I can get the stickies off Grovelbot’s board, but I can’t figure out how.

The text stickies are not images, but the image ones you can copy their URL and paste it into your sig via your Profile page, then it will appear in your sig at the bottom of all your posts.

Just go to the My DU tab, then click on Profile, then Edit Signature Line. Paste the image URL there, and then Update!

If you're using Windows you can Right-Click on an image and Copy Image Link

Or am I misunderstanding what you are trying to do?

Or a smilies like this one: