There's no right way to do Bloom because it's basically an hack, so you do it anyway you want to achieve the result you want ;)

In my opinion using FP16 render targets in the bright/blur passes produces better results because you might get some color information during the blur that would be clipped in the bright pass if you used 8-bit render targets.

There's no right way to do Bloom because it's basically an hack

Bloom is no hack. It's actually just a part of the flares lenses produce. Because no glass is perfect a really small amount of light always gets refracted onto nearby pixels. Normal distribution (aka gaussian filter) doesn't produce the most physically correct results though. An exponential distribution is better suited for Bloom, but it's not separable thus being not well suited for a realtime application. You can approximate an exponential distribution by using multiple gaussians though which is what they are doing in CryEngine 3 and Unreal Engine 4.
Also it doesn't really make any sense to implement a bright pass to subtract a certain amount of light so that only the bright light gets blurred. That's often a sign that the engine is not actually High Dynamic Range. Light always scatters onto surrounding pixels, just a tiny amount of it, but it's completely independent of its brightness. So a proper exponential filter and maybe a multiplicative factor should be about it.

BTT: If you want to have High Dynamic Range Rendering, you should keep the Bloom in High Dynamic Range and later perform Tone Mapping to resolve it to Low Dynamic Range. Also there's no need for an actual bright pass. Just do that in the Bloom Pass to save Bandwidth.

Also a cool trick is to use a R10G10B10A2 resource as a backbuffer. Most monitors can actually display this when the application runs in full screen. So you can actually have a higher dynamic range than the usual B8G8R8A8 backbuffer.