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Ah, yes, the SoundCloud logo...

It is available as an SVG via wikipedia. It has an orange gradient, which for the attached file I made white.

I also added a bleed amount as the background...

Yes, delete the black rectangle. Draw it in AI.

As for the GIF, it, being a bitmap, will never look good above the 100% of its native size.

If you upload the GIF, it would help advise.

For your background, delete the black bitmap and draw the path (a square rectangle) in AI.

I have no idea what all the little paths are for, so if you don't need them, delete them via the layers...