The security proof for SNARKs requires constant-depth compliance predicates, otherwise the proof doesn’t follow because the extractor size could blow up. I would have to read the coda whitepaper to see what their assumptions are for their security proof. I was just curious whether the equivalent of a birthday attack would be possible for some state transition if the depth was increased, since I’m not aware of a formal proof of soundness (and I heard in passing that recursion depth would affect security properties of nested snarks, but hearsay!)

I don't think circuit depth matters for SNARKs? Though there are other proof systems where AND depth matters.

Also, though I haven't read the CODA paper either, I think they would only need a single fixed circuit containing a SNARK verifier. Since for a given security level, SNARK sizes and verification steps are constant.