we could say that a Global Brain already exists

Yes, we could... and it has been vehemently debated. There are aspects of our global consciousness that are "Global Brain-like" - but IMO what Global Brain will be is an environment where we are immersed in intelligence. And our environment definitely isn't that yet. And we would be much more well connected and intimately connected with our minds via the substrate of the Internet. And that hasn't happened yet either. But it is a good debate - I feel that at this point is productive to continue having it.

Also, I haven't read Hofstadter but I'll make sure to before October.

what measures you might use for calculating the evolution (or emergence) of a GB

This is still debated. In one of the first papers on Global Brain, Mayer-Kress suggests that density and quality of agent connections will be more important than absolute intelligent agents in the system. I tend to agree. We could model the density and quality of connections in the brain to figure out what type of system we would need to produce on our scale of reality.

to what extent will the GB depend upon our higher intelligence, and to what extent will it exist independently of us?

I think everything in the universe is produced bottom-up... so ultimately it would be a higher intelligence produced by our collective interactions. I'm pretty sure.

Does this include a singularity scenario?

I believe that, although I question the name singularity, what the singularity describes is the neurons more deep connection with the Internet substrate - so they are mutually supportive ideas. I will go into great detail about it in the part 5.

Finally, I'm not sure I agree that only crystals exhibit non-living, non-random behavior. I think by that measure you could say that all objects exhibit non-random behavior in their gravitational interactions. You could probably extend that to the atomic forces, quantum systems, rivers, etc. There are lots of non-living processes where entropy is reduced within the observed system, but increased overall.

Don't have time to respond to this now - but you have given me a lot of reason to reconsider - I want to elaborate on what I meant further to see if there is any merit to it.

Thanks for your response :) Very helpful!

EDIT:

So in regards to your comment about my analysis of living/non-living. I'm glad you brought it up. I think what I am referring to, or the that inspired me to claims this, is autopoiesis. Life "creates" internal order from the surrounding environment. In this way they anti-entropic. I don't feel any non-life creates internal order, they are just drifting towards disorder. Is this a fair assessment? I am not completely wedded to the idea, I just wanted to propose it. What do you think?