Species identification: what kind of fungus is this?
I think these are parasitized insects.  

Fungi of various types can infect insects.  The end stage produces these fruiting bodies.  Here is an example I found on the excellent site WhatsThatBug.



Here is another example.



http://www.anura.it/the-parasites-parasite-everyone-has-a-bad-day/clasp04/

The second site notes


  Entomopathogenic fungus (Clavicipitaceae sp.) killed a troglophile
  moth, Veins cave (Grotta delle Vene), Italy


I would guess something similar for what you found. 

I must say your images are incredible.  The long fungal tendrils on the second image are the most amazing example of this I have ever seen.