In general, yes, there will often be a performance benefit to using "thinner" (fewer bytes per pixel) FBO formats. It depends on the details of your hardware and how you're using the buffers, but it's quite common for performance to be memory-bandwidth-limited for at least part of the graphics frame (postprocessing, for instance). In that case, a thinner FBO format means a smaller amount of memory to access, which means it'll be faster.
Assuming you only need the extra precision for the alpha channel, but not the color channels, it's also possible on recent GPUs to use multiple render targets (MRT) with an 8-bit format for the RGB and a second, separate render target in single-channel R16F or R32F format for the alpha. (Your shaders would have to be altered to output the alpha to the second render target.) That might be a performance gain too, especially if you need to perform postprocessing operations on the alpha channel alone - it'll be faster to read and write if the alpha is in a buffer by itself.
As for which formats are supported, any GPU supporting Direct3D 10 or 11 (which is probably most of the discrete desktop GPUs sold in the last 5 years, but not integrated or mobile GPUs) should support RGBA16, RGBA16F, and RGBA32F. However, the oldest GPUs within that 5-year time frame might not support bilinear filtering or mipmaps for RGBA32F, although they should support it for RGBA16F.
All in all, RGBA8 is generally going to be the fastest format, but if RGBA8 isn't doing it for you due to precision or range problems then I'd recommend trying out RGBA16F. It has broader support than RGBA32F, is quite possibly faster as well, and probably gives plenty of precision for whatever you're doing.