You can key it, you can mask out the white or you can use rotobrush on it.
AE does not support palette-based transparency. Export the GIF as an image sequence from Photoshop.
After Effects is lousy at creating animated gifs. If you need to do the animation in AE then render a lossless with alpha movie and open that in Photoshop and use the save for web feature to create your animated gif. If you are trying to fix an animated gif that has a problem then open that gif in Photoshop and do your clean up there.