This honestly wouldn’t have anything to do with the theme. All that’s happening in regards to the theme is that you’re putting in am image URL to display for the logo and it’s displaying that. This would all have to do with your image and the process of uploading it through WordPress.
I’m honestly not sure why uploading a GIF through WordPress, and it’s image processing, would cut out the GIF animation. The only thing I can think of is you’re accidentally using one of the thumbnail sizes that WordPress generates from your uploaded image, opposed to the actual image you uploaded.– If you were to provide a working URL to your site, I could tell you if you’re doing that or not; right now, I see a coming soon page.
However, the low quality with transparency showing through to a non-white background is expected, as a GIF is the lowest possible quality image type you could use. If you wanted a nice, transparent image, you’d need to use a PNG. However, that would obviously not allow for any kind of animation. This is honestly a bit old-fashioned, though; you may want to consider ditching that aspect.