You don't need:
in your .vmt. The "$bumpmap" is all you need. Your animation proxy is correct, but did you actually compile your normalmap as an animated texture?
You can simply use DXT1 compression for normal maps without nice filtering. If you are using the alpha channel of the bumpmap then use DXT5. Uncompressed textures are much larger and rarely necessary to use.
You are correct that the DIFFUSE image is the visible part of your material, 99% of the time the diffuse image is simply your basetexture. BUMP and NORMAL are the same thing in source and are created using either CrazyBump, nVidia NormalMapFilter or whatever gimps is called.
It actually doesnt really matter if the textures are the same size or not, smaller textures will be automatically "scaled" up so they are the same size as the larger ones.
Generally bumpmaps are not that often used on transparent textures, and depending what engine version you are using adding $transparent 1 to your .vmt may be stopping the bumpmap from working - I'm honestly not sure what works in which versions anymore.
The wiki is definitely a good source of information, but be careful sometimes as the engine undergoes updates and changes however the wikis arent always changed to keep up with the new info.
In the end I recomend you making 2 textures... the first is just you basetexture (diffuse) from your material using lightmappedgeneric or unlitgeneric. The second make into a refract texture using your normalmap as the animated refract. You can then simply create 2 surfaces, one over the other and put one texture on each.
*you replied while i was writing this so the info may be a little inconsistant with the last 2 posts, but ill leave it anyway.