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[The Fifth Horseman]

The best idea for saving the image is using GIMP - palette mode PNG's are usually quite small and at the same time razor-sharp. That made my avatar under 6 kb.

To change your avatar, you need to go to your Avatar Controls. Max dimensions are 100x100, max filesize is 30 Kb. That's a fair lot, I assure you.
The "Enter a URL to an online avatar image" is used to direct-link to files stored offsite. Use that only when your avatar is stored at your own webspace or photo album (say, on Photobucket.com).
"Upload a new image from your computer" is what you are looking for, and will store your avatar on Abandonia, copying the selected file directly from your hard drive.