The question is in the title: which different image/animation file formats can be embedded here? I have searched around without finding any list of formats.
Not a complete list.
These definitely work:
PNG, JPEG, GIF
Pretty sure these won't work (currently at least) SVG
I haven't spotted a bmp or a tiff yet (and not certain stack.imgur.com supports those), but there might be ways to do them. However, the fact that on stackoverflow many posts discuss BMP files but display PNG files when talking about them (e.g. like this) suggests that maybe BMP files are either not possible or may be possible but might be actively discouraged (if so, presumably for space reasons); if that's the case TIFF files would have similar issues.
There are other options that work, albeit not via imgur
Basically anything that works with the html tag
<img> but this is browser dependent