Background:
As shown on the screenshots the background is the level zero layer (the black one). Uncheck it to have transparent background. Gif should be ok then...
To save:
First :
Always save in .pdn format
AND second :
save in any other format you need...(.jpg, .gif, etc)
(somethime you'll need to flatten before Ctrl+Shift+F)
If you need to update/change then open the .pdn file!