Q: Shankara often wrote the descriptor “pure Consciousness” to point to Brahman.
1. What does “pure Consciousness” have to do with conventional consciousness, as in “I’m conscious of this or that?” Does chidabhasa explain it?
A: chidAbhAsa is the best metaphor, I think, (it is pratibimba vAda and associated with vivaraNa). The other main one is avachCheda vAda, associated with bhAmatI, which uses the idea of upAdhi-s. Consciousness (big ‘C’) is typically used to refer to non-dual reality; ‘c’onsciousness is the manifestation of ‘C’onsciousness in the mind of man.
2. Is there a difference between Consciousness (as-if paramartha level) and existence?
A: As you know (!) you cannot define or say anything objective about Consciousness. Ideally you should read the long Shankara commentary on satyam j~nAnam anantam brahma in Taittiriya Upanishad 2.1. That explains how such ‘descriptions’ work. The adjectives qualify-support-limit each other so that you do not take any single one as in any way a descriptive attribute. If you want, you could say that Brahman is limitless-existence-consciousness. But at the pAramArthika level, you cannot say anything at all about Brahman!
3. If there is a difference, which is more fundamental: Consciousness or existence? I.e. which gives rise to which? Why (not the other way)?
A: I cannot really add anything to the previous answer.