One problem is that your circle ends at 0, as does the rest of the texture. When it reaches, max, it will fill everything around it as well.
This should help:

You want this one:
Graphics.Blit(Texture, RenderTexture, Material, int);

1. Get the current screen contents into a texture, probably a render texture, called sceneContent
2. Blit into a RenderTexture, called hBlurredRT, using the horizontal blur material, which is using the horizontal blur shader
3. Blit again, into another texture called finalRT, using a material that has the vertical blur, having passed in the hBlurred RT

RenderTexture sceneContent; // RT with scene content
RenderTexture hBlurredRT; // RT with the scene, horizontally blurred
RenderTexture finalRT; // Final render, draw this to the screen after blurring

Material hBlur; // Horizontal blur material
Material vBlur; // Vertical blur material

Then, when you want to render:
Graphics.Blit(sceneContent, hBlurredRT, hBlur, 0); // Blur sceneContent with hBlur. Store the output in hBlurredRT
Graphics.Blit(hBlurredRT, finalRT, vBlur, 0); // Blur hBlurredRT vertically, using vBlur. Store in finalRT

Note that these assume that your main input texture is internally named _MainTex

A texture contains data, it's the same thing in this case :)

Just subtract 0.5 before plugging it into Alpha clip. However, using the full range, as in partial transparency, means you need to use proper alpha, not just alpha clip, as alpha clip is a binary operation

It is possible, but if you look at the site, there are two shaders, not one :)
So, you can make both of these shaders, render the first one into a Render Texture, then pass that render texture (RTBlurH) into the second shader.

This has been fixed in 1.05

This issue has been fixed in 1.05

That's the naive solution, yes. It would work in a specific case, but it wouldn't solve all cases. You need to branch based on:

1. Whether or not lightmapping is enabled on the object
2. If you're using deferred or forward
3. Which lightmap type you're using. Single vs Dual vs Directional
4. And, for me, whether or not you're using Unity 5

Good!