Does premultiplied alpha give order independent transparency?
Premultiplied alpha itself does not give you order independent transparency, no.

This page talks about how it can be used as part of an order independent transparency solution however: http://casual-effects.blogspot.com/2015/03/implemented-weighted-blended-order.html

Other benefits of premultiplied alpha include:


better mipmaps for textures that contain alpha: https://developer.nvidia.com/content/alpha-blending-pre-or-not-pre
Being able to do both transparency and additive blending within a single texture without needing to change render states or issue multiple draw calls: http://amindforeverprogramming.blogspot.com/2013/07/why-alpha-premultiplied-colour-blending.html
Being able to represent more realistic lighting situations, such as the fact that being non opaque doesn't always mean transparency: http://distantsoulsdev.blogspot.com.ar/2013/02/premultiplied-alpha-online-interactive.html
Being able to more naturally / correctly / effortlessly composite images that have alpha channels.
Being able to get better results from image processing (blurs / convolution). No black halo: https://stackoverflow.com/questions/4854839/how-to-use-pre-multiplied-during-image-convolution-to-solve-alpha-bleed-problem
Premultiplied alpha also allows transparent objects to be drawn front to back instead of back to front, possibly saving pixel writes.