Is Eywa a fungal organism?

[Human No More]

Zooplantae actually, but yes.
There are certainly fungi on Pandora (remember that giant mushroom that Jake lands on? ) and those are going to be part of Eywa like all the other plant species are... But in a way, I suppose the overall structure could be seen as very similar, it's just that fungi on Earth don't have the size or biological sophistication for consciousness.

I think that a 'Gaia' is possible, but doesn't actually exist (on Earth, anyway)0. After all, as far as I understand the idea, it works in a very similar way.

As for the method of signal transduction, I don't think it is unobtainium as there is already a method of transferring electrical signals via neurons - I'd guess that the roots simply include similar structures and act as synapses.